Computational study of statically confined electron systems by means of quantum chemical and machine learning techniques

DOCTORAL DISSERTATION Computational study of statically confined electron systems by means of quantum chemical and machine learning techniques/ Estakikoki Konfinatutako Elektroi Sistemen Azterketa Konputazionala Machine Learning eta Kimika Kuantikoko Metodoak Erabiliz XABIER TELLERIA ALLIKA Supervised by Dr. Jon Mattin Matxain Beraza and Dr. Jose Maria Mercero Larraza OCTOBER 2022 Computational study of statically confined electron systems by means of quantum chemical and machine learning techniques/ Estakikoki Konfinatutako Elektroi Sistemen Azterketa Konputazionala Machine Learning eta Kimika Kuantikoko Metodoak Erabiliz Dissertation presented in Submitted in fulfillment of the requirements of the D O C T O R A L P RO G R A M I N T H E O R E T I C A L C H E M I S T RY A N D C O M P U TAT I O N A L M O D E L L I N G Presented by X A B I E R T E L L E R I A A L L I K A Supervised by D R . J O N M AT T I N M AT X A I N B E R A Z A and D R . J O S E M A R I A M E R C E R O L A R R A Z A Euskal Herriko Unibertsitatea/Universidad del Pa´ıs Vasco & Donostia International Physics Center In DONOSTIA,OCTOBER 2022 (cc) 2022 Xabier Telleria Allika (cc by-nc-nd 4.0) CONTENTS Contents iii List of Figures vi List of Tables ix 1 introduction 1 1.1 The electronic structure problem . . . . . . . . . . . . . . . . . . . . 1 1.2 Importance of electron correlation . . . . . . . . . . . . . . . . . . . 3 1.2.1 Wigner crystals and molecules . . . . . . . . . . . . . . . . . 3 1.2.2 Quantum Dots and Hooke atoms . . . . . . . . . . . . . . . . 4 1.3 Machine Learning assistance . . . . . . . . . . . . . . . . . . . . . . . 5 1.3.1 Supervised learning: neural networks for prediction . . . . . . 6 1.3.2 Semisupervised learning . . . . . . . . . . . . . . . . . . . . . 7 1.4 Scopeofthiswork ............................ 9 2 quasi-one dimensional systems 11 2.1 Introduction................................ 12 2.2 Computational protocol . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.1 Quantum mechanical model . . . . . . . . . . . . . . . . . . . 12 2.2.2 Distributed Gaussian orbitals . . . . . . . . . . . . . . . . . . 13 2.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.1 Two electron system . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.2 Systems of three and four electrons . . . . . . . . . . . . . . . 20 2.4 Concludingremarks ........................... 23 3 quasi-two dimensional systems 25 3.1 Introduction................................ 26 3.2 Computational protocol . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2.1 Distributed Gaussian orbitals . . . . . . . . . . . . . . . . . . 27 3.2.2 Quantum mechanical computations . . . . . . . . . . . . . . . 28 3.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3.1 Oneparticlesystems....................... 29 3.3.2 Few particle systems . . . . . . . . . . . . . . . . . . . . . . . 32 3.4 Concludingremarks ........................... 35 4 spherical hooke atoms 37 4.1 Introduction................................ 38 4.2 Computationalmethods ......................... 40 4.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.3.1 Two-electron systems . . . . . . . . . . . . . . . . . . . . . . . 42 4.3.2 Four-electron systems . . . . . . . . . . . . . . . . . . . . . . 44 4.3.3 Six-electron systems . . . . . . . . . . . . . . . . . . . . . . . 45 iii iv contents 4.3.4 Eight-electron systems . . . . . . . . . . . . . . . . . . . . . . 45 4.3.5 Ten-electron systems . . . . . . . . . . . . . . . . . . . . . . . 46 4.3.6 Screened Hooke Atom . . . . . . . . . . . . . . . . . . . . . . 47 4.3.7 Decomposing the energy into different contributions . . . . . 48 4.3.8 CoulombHoles.......................... 51 4.4 Concludingremarks ........................... 53 5 ml assisted phase diagrams 61 5.1 Introduction................................ 62 5.2 Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.2.1 Thermodynamic model for eutectic composition . . . . . . . . 63 5.2.2 Quantum chemical methods . . . . . . . . . . . . . . . . . . . 64 5.2.3 Semi-supervised learning techniques . . . . . . . . . . . . . . 66 5.2.4 Work procedure and calculation setup . . . . . . . . . . . . . 67 5.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.3.1 Solid-liquid phase diagrams with a single eutectic point . . . 69 5.3.2 Ground state spin multiplicity for Hooke atoms . . . . . . . . 71 5.3.3 Detecting covalent bonding in spherically confined He2 systems 73 5.4 ConcludingRemarks........................... 76 6 gaussian confinement and connection to hooke atoms 77 6.1 Introduction................................ 78 6.2 Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . 78 6.2.1 One-body integrals concerning Gaussian confinements . . . . 79 6.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 82 6.3.1 Deeply confined one centre systems with even number of electrons 82 6.3.2 Loosely confined two electron systems with screened Coulomb interaction ............................ 84 6.4 ConcludingRemarks........................... 88 7 summary, main conclusions and future works 89 7.1 Mainconclusions............................. 90 7.2 Furtherworks............................... 91 7.2.1 Some preliminary results concerning endohedral systems . . . 92 8 euskarazko laburpena 95 8.1 Sarrera................................... 95 8.1.1 Egitura elektronikoaren problema . . . . . . . . . . . . . . . . 95 8.1.2 Korrelazio elektronikoaren garrantzia . . . . . . . . . . . . . . 97 8.1.3 Ikasketa automatikoaren laguntza . . . . . . . . . . . . . . . 99 8.2 Lanhonenhelburua ........................... 102 8.3 Egindako lanen laburpenak . . . . . . . . . . . . . . . . . . . . . . . 103 8.3.1 Sasi dimentsio bateko sistemak . . . . . . . . . . . . . . . . . 103 8.3.2 Sasi bi dimentsio sistemak . . . . . . . . . . . . . . . . . . . . 106 8.3.3 Hiru dimentsioko Hooke-n atomoa . . . . . . . . . . . . . . . 108 8.3.4 Makina ikaskuntza bidez lorturiko fase diagramak . . . . . . 113 contents v 8.3.5 Potentzial gausstarrak . . . . . . . . . . . . . . . . . . . . . . 117 8.4 Emaitzanagusiak............................. 120 8.4.1 Konfinamendu harmonikoa ikertzeko protokolo konputazionalen garapena ............................. 120 8.4.2 Ziurgabetasun laginketen metodoen inplementazioa fase diagramaklortzeko ......................... 120 8.4.3 Konfinamendu gausstarrei dagozkien gorputz bateko integralak inplementatu eta aurretiaz lorturiko emaitzak hedatu . . . . 121 8.5 Ondorionagusiak............................. 123 8.6 Etorkizunerakolanak........................... 124 bibliography 127 LIST OF FIGURES Figure 2.1 Normalised density profiles for singlet (blue) and triplet (orange) spin states for several confinement strengths k . Notice in the weak confinement limit, both profiles are alike and they differ in the strong confinement limit. . . . . . . . . . . 14 Figure 2.2 Variation of the location parameter with respect to the confinement parameter k for singlet and triplet states, the orange square represents the asymptotic limit for location index of thetripletstate.......................... 15 Figure 2.3 Particle-hole entropy for two electron system in high spin state for several values of confinement parameter k . Maximal entropy value is found for a value around k=1×10−3.3 . . 16 Figure 2.4 Energies for singlet and triplet spin states using minimal gaussian basis (left) and energy difference between the singlet and the triplet states for the two electron system using minimal basis (right). The zero energy gap happens around a value close to k=5×10−4. ................. 20 Figure 2.5 Normalised density profiles for three (blue) and four (orange) electron systems in high spin configuration for several confinement strengths k. .................... 21 Figure 2.6 Computed electron-hole entropies for n= 3, 4 number of electrons with high spin state for several values of the confinement parameter k. .......................... 22 Figure 3.1 Geometry of the distributed gaussian functions according to g number. All distances between closest neighbours is given by the parameter δ........................ 27 Figure 3.2 Logarithm of absolute errors (a) and standard deviations (b) for the ten first states and the first three degenerate groups with ( nx+ny) = { 1, 2, 3 } . Same color state represent analytically degenerate states. . . . . . . . . . . . . . . . . . 30 Figure 3.3 Obtained estimated relative errors for estimated δopt using various NN models consisting of two hidden layers with several number of neurons per layer. . . . . . . . . . . . . . . . . . . 32 Figure 3.4 Density difference between the CASSCF(2,8) and HF singlet spin wave-functions as a function of the distance with respect to the potential minimum for several values of k. ...... 34 vi LIST OF FIGURES vii Figure 3.5 Density difference between the CASSCF(3,8) and HF quartet spin wave-functions as a function of the distance with respect to the potential minimum for several values of k. ...... 34 Figure 4.1 Triplet-singlet energy gap, in eV, calculated at the CAS( Ne , Norb ) (dashed line) and MRMP2( Ne , Norb ) (continuous line) levels of theory with the ETBS-6S basis set, as a function of the number of orbitals Norb included in the active space. All cases correspond to a CASSCF wavefunction in which all electrons are included in the active space, except for the curves in magenta for the 10 electron system, that correspond to wavefunctions in which the 1s orbital occupation is set to 2, and therefore, the active space is composed of 8 electrons and Norb-2orbitals. ......................... 49 Figure 4.2 A) Left Figure: Triplet-singlet energy gap, in eV, calculated at the MRMP2( Ne ,10) level of theory (2-, 4-, 6-, and 8electron systems) and MRMP2(8,14) (10-electron system), as a function of the number of electrons and for different values of λ . B) Right Figure: Triplet-singlet energy gap, in eV, calculated at CAS( Ne ,10) (dashed line) and MRMP2( Ne ,10) (continous line) levels of theory for 2,4,6 and 8 electrons, and CAS(8,14) (dashed line) and MRMP2(8,14) (continuous line) levels of theory for the 10-electron system, as a function of the degree of screening ( λ ). All calculations were done with the ETBS-6S basis set. . . . . . . . . . . . . . . . . . . . . . 49 Figure 4.3 Coulomb holes calculated at the CASSCF( Ne ,13)/ETBS6S level of theory for both full (solid lines) and λee = 1.0 screened-coulombic (dasshed lines) Hooke systems with different number of electrons ( Ne =2,4,6,8 and 10) and for singlet(black lines) and triplet states (red lines). . . . . . . . . 52 Figure 5.1 Decision tree used for labelling each point in the (x1 , T) plane using eutectic point coordinates and melting curves. . . . . . 64 Figure 5.2 Flowchart for the general procedure for phase diagram construction.............................. 68 Figure 5.3 Ag/Si solid-liquid phase diagram. Green is the heterogeneous solid phase α+β , dark blue is the homogeneous ideal liquid mixture L , light blue is β+L and purple is α+L ; the white dots represent the sampled points. . . . . . . . . . . . . . . . 69 Figure 5.4 KNO 3 /LiNO 3 solid-liquid phase diagram. Green is the heterogeneous solid phase α+β , dark blue is the homogeneous ideal liquid mixture L , light blue is β+L and purple is α+L ; the white dots represent the sampled points. . . . . . . . . . 70 4introduction a substrate for Wigner crystals seem to be bilayer materials in which low densities are obtained by means of emerging moir´e patterns [28–31]. In the context of non-periodic finite systems, localized electronic states are also called Wigner molecules and they emerge when the confinement of the electrons is rather weak [32–34]. Thus, in order to describe Wigner molecules or, in general, low density electron systems, electron correlation effects on both, energy and wavefunction, must be treated by means of high enough computational methods. Highly accurate densities for systems in low-correlation regimes as well as in high-correlation regimes have been obtained in the literature [35–38]. If similar systems containing larger number of electrons ( n > 2) are to be considered, a richer variety of electronic states consisting on several ground state spin multiplicities and multi-determinantal features arise [39–41]. Although such systems can be employed to understand manybody interactions, the computational cost increases with the size of the system. In spite of these difficulties, these model systems have been repeatedly used in the calibration of electronic structure methods, for they provide very variable dynamic and nondynamic electron correlation regimes [42–44] that pose a great challenge for current computational methods. [37, 45–57] Such calibration has been possible because of the recent availability of highly accurate analytical and benchmark data.[39, 40, 58–64] 1.2.2 Quantum Dots and Hooke atoms Quantum Dots (QDs) are regions of space where, due to a given external potential, particles such as electrons are confined. As these regions shrink, a critical size is reached at which properties of the microscopic world - as energy quantisation, electronic and magnetic structure - emerge. From a physical chemical point of view, a molecule or a nanoparticle that confines its valence electrons can also be taken as QDs. Nowadays, in the dawn of quantum technologies, QDs form a vast and interesting field of study. These systems constitute one of the most elemental components for nanoscale devices which have a wide range of applications such as: solar energy harvesting, optoelectronics and quantum computing devices among many others [65]. Very recent works show further fields of applications for QDs as thermoelectricity [66], catalysis [67–70] and, most remarkably, quantum computation [71–75]. One of the simplest and most adequate models used in theoretical studies concerning QDs are the so called Harmonium or Hooke’s atoms in which electrons are confined in a spherical harmonic potential [76] for which the Hamiltonian is given as in equation (1.4) . Such models contain parameters that may be tuned in order to represent features corresponding to real QDs [77, 78]. For instance, work carried out in our group using a Hookean exact three-body model to examine electron correlation in a two-electron spherical quantum dot confirmed that triplet-singlet transitions take place as the externally applied magnetic field increases[79]. However, 1.3 machine learning assistance 5 the limitation of using an exact model restricted our study to two-electron systems. That is, the analytical solutions[80] for specific curvature parameter of the twoelectron Hooke atoms ( ω2=1 4 , 1 100 . . . ) are well known, which can lead in principle to highly accurate densities for systems in low-correlation regimes ( ω2→ ∞ ) as well as in high-correlation regimes ( ω2→ 0) [35–38]. If similar systems containing larger number of electrons ( n > 2) are to be considered, a richer variety of electronic states consisting on several ground state spin multiplicities and non-dynamical electron correlation (multi-determinantal features) arise [39–41]. Although such systems can be employed to understand many-body interactions, the computational cost increases with the size of the system. H=−1 2 n X i=1∇2 i+ω2 2 n X i=1 r2 i+ n X j>i=1 1 rij (1.4) As in Hooke model atoms, the incorporation of electron correlation effects has been shown to be essential for an adequate interpretation of the experimental spectra and transport properties in Quantum Dots [81–86]. In quantum dots, as opposed to real atoms, the effect of electron correlation may be varied at will through manipulation of the dimension and shape of the nanocrystal as well as of the strength, boundaries and symmetries of the confining fields[87, 88]. Besides, the electron-electron interaction can be screened due to lattice, the doping or the charges induced on the metal gates[89]. This fact makes the quantum dot many-body problem more complex than the more familiar atomic case. Finally, Hooke model systems have been repeatedly used in the calibration of electronic structure methods, for they provide very variable dynamic and nondynamic electron correlation regimes [42–44] that pose a great challenge for current computational methods. [37, 45–57]. Such calibration has been possible because of the recent availability of highly accurate analytical and benchmark data [39, 40, 58–63]. 1.3 MACHINE LEARNING ASSISTANCE IBM defines Machine Learning (ML) in the following way ”Machine learning is a branch of Artificial Intelligence (AI) and computer science which focuses on the use of data and algorithms to imitate the way that humans learn, gradually improving its accuracy”. According to this definition, any ML solution is based on two pillars: the data (which are empirically obtained abstract numbers) and an algorithm which operates upon these data in order to gain some kind of knowledge. In addition (as it is implicitly stated), employing a given algorithm, the quality of such knowledge improves as the number of data increases. 6introduction Machine learning methods are recently being employed in various fields of chemistry [90–93] such as: electronic structure theory development [94–98], efficient new phase discovery and phase diagram building [99–102], patterns in chemical properties [91, 103–106], obtaining Potential Energy Surfaces [107–111] and reaction paths and outcomes [112–114] among others. Taking this brief compilation of examples into account, ML methods are and will be important tools in chemical sciences. From a wide perspective, there are two ’traditional’ machine learning disciplines, the so called unsupervised and supervised learning paradigms. Given a set of data X={x1. . . xn} the former aims to find interesting structure in the data such as clustering and dimension reduction by techniques such as k-means Clustering, and Principal Component analysis. Meanwhile, the latter also considers one more set Y={y1. . . yn} and its goal is to learn a mapping which connects both sets X and Y by techniques such as Linear and Logistic Regression, Random Forests and Vector Supported Machines. Then, semi-supervised learning is halfway between these two; while still considering both X and Y sets, the number of elements in X is far larger than the number of elements in Y and thus any obtained mapping function turns out to be inaccurate [115–117]. In this work two supervised learning algorithms have been employed, i.e. Random Forests and Neural Networks as well as one semisupervised algorithms: Uncertainty Sampling. 1.3.1 Supervised learning: neural networks for prediction As it has been stated in the previous section, supervised learning techniques aim to build a machine f:X→Y which is able to map a feature set X into a label set Y such that (bearing in mind the general ML definition) it gets better at doing so by increasing the size of both sets. Given a labelled data set D , a proper ML algorithm is able to infer the required function f form training data contained in the set T ⊂ D and an error for such inferred function can be obtained by using the complementary set E ⊂ D , T ∩E =∅ . Among the thousands of techniques designed for such purpose, in this work deep Neural Networks (NN) have been employed to infer optimal basis set parameters as it is shown in chapter 2. In the mid of the XX century, with the aim of modelling how human brains process information, the works of W. S. McCulloch, W. Pitts and F. Rosenblatt gave rise to the first implementations of the perceptron [118, 119] which by means of non linear functions (as the sigmoid [120]) operating upon weighted sums of input gave certain logical or continuous outputs. As it has been proved, by arranging a given number of these neurons in layers and coupling these several layers such that the output of the i -th layer becomes the input of the (i+ 1 ) -th one, these machines are (in principle) universal approximates [121–125]. That is, given enough data and large enough number of layers and neurons in each layer, these networks can be trained to approximate any arbitrary function. 1.3 machine learning assistance 7 1.3.2 Semisupervised learning Most supervised machine learning prediction and classification models work provided the extension of the learning data set is large enough. Nevertheless, in many real-world situations, as in our case, obtaining independent variable values (labels) may be time-demanding or computationally/economically expensive. Thus, it is wise to gain further expertise based on already established available knowledge and invest our resources (will, time and money) in enlightening the darkest corners of our ignorance. Semi-supervised and active learning techniques can be applied upon data for which we have two different sets: a labelled set L and an unlabelled set U . Namely, the labeled set L is composed by points formed as an ordered Cartesian product (x1 , y1). . . (xl , yl) where YL={y1. . . yl}∈{ 1 . . . C} is the set of possible labels (categories). Then, we also have the unlabelled set U composed by (xl+1 , yl+1). . . (xl+u , yl+u) points where the labels YU={yl+1. . . yl+u} ∈ { 1 . . . C} are unknown and the cardinality of this set is much larger than the former one, l << u. By doing so, we shall know where to map the next experiment to obtain phase diagrams as it has been done in very recent works [126, 127]. We have been using two complementary techniques: label propagation and uncertainty sampling. Under the initial assumptions, we have built a grid in parameter space with X={x1. . . xl+u} coordinates from which a number l will be labelled with just one label form categories Y={ 1 . . . C} and the remaining u coordinates are not labelled (of course, all categories are represented in the l labelled data, so card(l)≥card(C) ). We also assume that the number of unlabelled coordinates is far larger than the labelled ones l << u . The point here is to use the information (labels and coordinates) of the l labelled data and infer the probabilities for unlabeled data to belong to each of the C possible categories and compute the uncertainty based on these probabilities. 1.3.2.1 Label Propagation Algorithm X. Zhu and Z. Ghahramani developed an algorithm named ‘label propagation algorithm’[128] in which information from labelled data is propagated to unlabelled data in a stochastic process by means of Markov chains. Here, one defines a graph containing all -labelled and unlabelleddata and defines the connection strength (weight) by pairs. Using properties related to homogeneous Markov chains and stochastic matrices, it can be shown that the algorithm converges to a unique stationary solution (1.5) for any which initial guess for the matrix Y(0) we may make. Yu= (I−˜ Tuu)−1˜ TulYl(1.5) 8introduction The main drawback of this method is the fact that for dense or large enough grids, the computation of the matrix (I−˜ Tuu)−1 is expensive or, in the worst case, the matrix is singular. This problem may be sidestepped by, instead of using the closed formula (8.5) taking an initial label vector and iterating it. Nevertheless, since convergence is reached after several iterations and large matrices are still involved, this algorithm is still costly for dense enough grids and this deficiency is more remarkable as the number of dimensions of the feature space increases. As an alternative, other, rather cheap, classifiers such as Random Forests (RF) can be used to solve the problem. Since RF do not have over fitting problems, using a large enough number of trees, the method is universal for all systems studied in this work. Therefore, the initialisation set is employed to train a RF model which will label the unlabelled points assigning them to each category with a given probability, that is, an equivalent approach as using label propagation. 1.3.2.2 Uncertainty sampling Learning is widening our knowledge by gaining more information where our ignorance or uncertainty is deepest. In order to learn in an efficient way, one should focus on one’s lack of expertise rather than what she/he already knows. Uncertainty sampling is a way of sampling just the points in which the uncertainty is the largest, but how can we measure uncertainty? Provided we have a probability distribution function PC(y|x) such that given a parameter vector x gives the probabilities of that point to belong to each of the categories y in the set { 1 . . . C} , we may define some uncertainty measures as: Least Confident (1.6) , Margin (1.7) and Shannon Entropy (1.8) . The first one maximizes the least likely (therefore the most uncertain) of the unlabelled data, the second one is more flexible than the former in the sense that it takes into account the two most likeliest prediction points. Finally, the Shannon entropy is a measure of information content; the less information a variable contains, the more we know about its nature [129]. uLC (x) = 1−max CP(C|x)(1.6) uMS (x) = 1−[P(C1|x)−P(C2|x)](1.7) uSE (x) = −X C P(C|x)log P(C|x)(1.8) 1.4 scope of this work 9 1.4 SCOPE OF THIS WORK Once the state of the art concerning Wigner molecules, Hooke atoms and some machine learning techniques has been described, in this section the main goals of the present work will be provided. We shall now list the mains goals of this work: 1. Optimize distributed gaussian basis functions for correctly describing the oneand two-dimensionally confined electronic systems by means of machine learning approaches (Neural Networks) for variable confinement parameter k=ω2 . Besides, set an optimal wavefunction based electronic structure method for obtaining accurate wavefunctions. Employ the settled protocole in order to study the Wigner location for small number of electrons n={ 2, 3, 4 } in the high spin state. 2. Optimize one-centre even tempered basis functions using classical optimization techniques (simplex and Newton-Raphson) for properly describing three dimensional spherical Hooke atoms composed by few number of electrons n={ 2, 4, 6, 8, 10 } and confinement parameter k=ω2= 1 / 4. Settle an optimal wavefunction based method for obtaining accurate energies for the lowest laying singlet and triplet states. 3. Implement and improve machine learning methods based on semi-supervised learning and uncertainty sampling for obtaining phase diagrams efficiently and provide some chemically relevant systems. 4. Calculate analytical one-body integrals corresponding to gaussian confinement potentials in terms of gaussian basis functions and implement them in GAMESS US. Provide some examples related to the previous sections. Chapter 2 QUASI-ONE DIMENSIONAL SYSTEMS In this chapter, localization properties of systems composed by few electrons confined in a quasi-one-dimensional harmonic potential have been studied by semi-analytical procedures for two electron systems, and by more robust quantum chemical approaches concerning two, three and four electron systems in high spin state. By using electronic-structure properties obtained by multi-reference methods such as the one-body density and the particle-hole entropy, we have been able to define a path that connects the corresponding Wigner molecules with the Fermi liquids by varying the harmonic-potential confinement parameter k . We conclude that the particle-hole entropy is a smooth function of the confinement parameter and connects the two limit cases, and shows a maximum value which position depends on the number of electrons. Submitted to JCP Uploaded to the arXiv http://arxiv.org/abs/2207.12014 11 12 quasi-one dimensional systems 2.1 INTRODUCTION In the previous chapter, some characteristics about the nature of the Wigner crystals and molecules have been introduced. Due to the intrinsic nature of such systems, in order to build models which represent those phases electron correlation effects must be properly described. Some previous studies tackled the issue of one-dimensional systems composed by few electrons ( n= 2, 3 and 4) and the correlation effects varying the length of the confinement box and the computation method comparing ROHF with FCI [130]. Far from being only theoretical models, systems containing these number of electrons have been observed experimentally [131, 132]. In this chapter, we have studied one-dimensional systems with ( n= 2, 3 and 4) electrons confined in harmonic potentials with several orders of magnitude for the confinement parameter k . In a first step, we have seen that singlet and triplet states for two electron systems have similar density profiles in the weak confinement regime ( k→ 0) while for large confinement regime ( k→ ∞ ) the density of each spin state converges to the free particle density. Then, for high spin states, for all number of electrons, we have computed the electron-hole entropy which is linked to correlation effects [133–135] and we have been able to connect two lowest entropy phases: the Wigner molecule and the Fermi liquid. These entropy curves depend on the number of electrons and the potential parameter k and, therefore, on the density of the system. In all cases we have observed there is a certain value of k for which the electron-hole entropy is maximal, hence static correlation effects play a big role. 2.2 COMPUTATIONAL PROTOCOL 2.2.1 Quantum mechanical model In this chapter, we have studied systems composed by n={ 2, 3, 4 } electrons which interact via Coulomb potentials among them and are confined in a one-dimensional harmonic well, hence the Hamiltonian of the system can be expressed as in equation (2.1). H=−1 2 n X i=1∇2 i+k 2 n X i=1 r2 i+ n X j>i=1 1 rij (2.1) As it is well known [136, 137], in the pure one-dimensional case the Coulomb operator is singular; in order to sidestep this problem, we shall solve the Schr¨odinger equation by using three dimensional basis functions distributed along one unique direction. The final energy is then obtained by subtracting the transverse components of the kinetic energy for each electron [138]. In this chapter, we will study the behavior of such quasi-one-dimensional systems. 2.3 results and discussion 13 2.2.2 Distributed Gaussian orbitals In order to properly describe our systems, we use a one dimensional grid of gaussian functions given by equation (2.2) where the position of each basis function is distributed along the xaxis Ri= (xi, 0, 0). ϕi(r;α,Ri) = 2α π3/4 exp(−α(r−Ri)2)(2.2) These gaussians have been set as follows: given a number of electrons n and a harmonic potential with curvature parameter k , the semiclassical turning points of the highest energy level are given as x0(n , k) = ±(2n+1)2 k1/4 . Using these points as well as the position of the minimum of the potential well ( x= 0) we place a basis function in each of them and we add m equidistant gaussians between the semi-classical turning points and 2 m more to each side of these points. In this setting, there is a total number of M= 3 + 4 m basis functions and the distance between consecutive basis is given by δ=2|x0| m+1 . Since all basis functions had the same exponent α , the overlap between neighbouring functions is given as S(α , δ) = exp(−αδ2/ 2 ) . One may also rewrite the last overlap function in terms of a dimensionless parameter ξ=αδ2 as S(ξ) = exp(−ξ/ 2 ) . This dimensionless parameter characterizes the resolution of the basis set which in our case has been chosen to be ξ= 1.0 [130, 138–141] . By doing so, the exponents are dependent on the confinement, the number of electrons and the number of basis functions as α=ξ(m+1)2 4x2 0(n,k). 2.3 RESULTS AND DISCUSSION 2.3.1 Two electron system As it has been introduced in the previous section, for each confinement strength parameter k , we have fixed the position of three gaussians: one where the confinement potential has its minimum ( x= 0) and one on each classical turning points ( x= ±x0(k , n) ). Once these points are settled, we have added 100 equidistant gaussians ( δ=x0 13 ) with the same exponent ( α=δ−2 ), 50 of them in between the two turning points and 25 at each side of them, giving rise to a total number of 103 basis functions. Using these basis, we have computed the singlet and triplet spin states at CASSCF level of theory using several numbers of active orbitals for several values of the confinement parameter k . After having run some initial tests, we have concluded that the optimal active space is the one obtained from CASSCF(2,5). By using this 20 quasi-one dimensional systems k as can be seen in figure 2.4. For small values of k , both states are very close in energy while the triplet state is a bit bellow the singlet state, but for some value around k= 5 × 10 −4 both lines cross and then the triplet state is higher in energy for all k . It is convenient to recall that for large k , the energy of the triplet state goes as E0T= 2 k1/2 while the energy of the singlet state goes as E0S=k1/2+ (2/π)1/2k1/4. Figure 2.4: Energies for singlet and triplet spin states using minimal gaussian basis (left) and energy difference between the singlet and the triplet states for the two electron system using minimal basis (right). The zero energy gap happens around a value close to k=5×10−4. Considering the minimal basis model, we have shown that below a particular small value of k , the triplet state (which represent the more localized system) is lower in energy than the singlet state (which represents the less localized system) and beyond k≈ 5 × 10 −4 we find the opposite situation. Thanks to this simple model, we have been able to determine the order of magnitude of k for which localization may happen. 2.3.2 Systems of three and four electrons Based on the results of the previous section concerning two electron systems, we may briefly study systems composed by three and four electrons and see if we obtain similar behaviours. Bearing in mind the observed properties of the two electron system, we have constrained ourselves to the high spin state upon which CASSCF( n , 2 n ) calculations (being n is the number of electrons) with the grid of gaussian basis functions stated in the previous sections have been performed. The obtained densities are shown in figure 2.5. Once again, we have computed the particle-hole entropies using the occupation numbers for each systems and the obtained results are graphically represented in figure 2.6. 2.3 results and discussion 21 As it can be seen, even though both curves have similar shape, the value of k for which the entropy is maximum depends on the number of electrons (the larger the number of electrons, the smaller the value of k at which the entropy reaches its maximum). We shall infer that the behaviour is similar to the two electron system and thus the conclusions are equivalent. In the light of these results, we shall define a kind of transition state taking place at the maximal entropy point S(kmax , n) such that connects the Wigner molecule and Fermi liquid states. The size and position of such transition state depends, of course, on the number of electrons. Figure 2.5: Normalised density profiles for three (blue) and four (orange) electron systems in high spin configuration for several confinement strengths k. 22 quasi-one dimensional systems Figure 2.6: Computed electron-hole entropies for n= 3, 4 number of electrons with high spin state for several values of the confinement parameter k. 2.4 concluding remarks 23 2.4 CONCLUDING REMARKS In this chapter, we have studied systems composed by two, three and four electrons confined in a quasi-one-dimensional harmonic potential characterised by the confinement parameter k. For the two electron system, we have obtained analytical formulations using a minimal basis set and a one-determinant wavefunction. In this way, we have been able to model the Wigner-molecule regime as a triplet with two identical gaussians basis functions and the Fermi liquid regime as a singlet with a single gaussian basis function. Using this scheme, we have minimized the energy of each system for several values of the confinement parameter k and have seen that a crossing between the two types of wavefunctions happens around k= 5 × 10 −4 . This indicates that, at this level of theory, some change in the nature of the wavefunction must take place around this value. Basis sets composed of a large number of overlapping gaussian functions were also used in order to approach the limit of a complete basis set. We performed a scan of electronic structure derived properties such as one-particle density and particle-hole entropy and both indices coincide that for k values around 1 × 10 −4 and 1 × 10 −3 a transition is taking place either because the location index for the singlet and the triplet state start to diverge or because the particle-hole entropy reaches a maximum. Anyhow, the order of magnitude of k at which the transition starts is the same as the one obtained by the simplified model. Extending the study to high-spin states concerning three and four electrons systems, we have observed the same behaviour. In the two limit cases, the density and the particle-hole entropies correspond to the ones expected for the Wigner molecule (k→ 0 ) and the Fermi liquid ( k→ ∞ ). By varying the confinement parameter k , we have computed a smooth path connecting these extreme states such that the particle-hole entropy reaches a maximum. For very small values of the confinement parameter, we have obtained some numerical instabilities and have not obtained reliable information on how the hole-particle entropy behaves in this regime. Further work concerning the study of this regime can be done in a close future. Chapter 3 QUASI-TWO DIMENSIONAL SYSTEMS In this chapter, the process for obtaining optimal gaussian basis sets distributed in a hexagonal grid for modelling harmonically confined two dimensional systems is described. As a first step, these basis functions are optimized by means of the variational principle for one-body systems and the obtained results are compared to analytical energies. As a result, the obtained numerical values for the energies are not only accurate with respect to the analytical ones, but also degeneracy is preserved. Using properly optimized neural networks, we have been able to obtain general optimal basis functions for arbitrary values of the confinement parameter k in the range (1 × 10 −9 , 1 × 10 1 ). The quality of the obtained basis functions increases with the number of functions and perform better for larger values of the confinement parameter k. In progress 25 26 quasi-two dimensional systems 3.1 INTRODUCTION A vastly employed practice in electronic structure simulations is to expand the corresponding many-body wave function in terms of one-electron basis functions. In the particular case of quantum chemistry, the most popular basis functions are gaussian functions sitting on each of the nuclei which compose the studied chemical system [142–144]. Besides, the use of floating gaussians includes some flexibility to better describe spread electronic density without the need of including an explicit nucleus [138, 145–149]. As it has been shown in previous works, distributed s -type gaussian orbitals are rather good basis functions for describing many-body fermionic systems [130, 138– 140]. This approach may be employed to study two dimensional quantum systems which have wide experimental and theoretical interest [150–153]. However, the scale of some systems is of the order of the nanometer, for which if a single centre is considered, linear dependency problems will happen. In order to set an example, experimentally, Quantum Dots based on Graphene (GQD) and Transition Metal Dichalcogenides (TMDs) have been synthesised in a range of sizes in the interval 1-7 nm [154–163]. Among them, GQD with visible light photo-luminescence have diameters around 1.0-2.5 nm [164, 165]. With respect to moir´e cells (as generalising Mott physics), the densities of such states are of the magnitude of 1 × 10 16e−/m2 [166–171] and for smaller densities, some author indicate the presence of Wigner crystallisation [172–175]. Since for such low densities (as it was firstly predicted by Wigner himself [20] and experimentally confirmed for 2D systems [176, 177]) the inter-electronic interaction energy is larger than the kinetic one, it is vital to describe the electron-electron interaction using accurate techniques [37, 39, 40, 42– 64]. Bearing this in mind, setting a quantum chemical approach to model such large systems is crucial. We shall test the validity of our approach to describe Wigner location in many electron systems. The final aim of this work is to obtain an optimal two-dimensional hexagonal mesh composed by s -type orbitals being α the exponent of such orbitals and δ the distance between closest neighbours. These two parameters are selected such that given a harmonic confinement strength k , the size of the mesh and a parameter ξ that indicates the overlap between neighbouring orbitals, one gets the minimal one-particle energy. With the aim of getting general tunable basis sets, an Artificial Neural Network has been trained to produce the optimal mesh. Once the basis set is optimised for given input parameters, we have used quantum chemical methods for computing properties concerning the electronic structure of harmonically confined systems of electrons. By doing so, all technologies employed in this field concerning high level methods such as the multi-reference ones can be applied in larger number of electrons. 3.2 computational protocol 27 3.2 COMPUTATIONAL PROTOCOL 3.2.1 Distributed Gaussian orbitals Hexagonal grids composed by 3D s -type Gaussian orbitals given as in equation (3.1) have been employed. The number of gaussian functions is given by M= 3 g2− 5 g+ 1 where g≥ 3 is related to the number of shells in the hexagonal pattern. In figure 3.1 a schematic representation of the geometry of the basis set is given for several values of gnumber. ϕi(r;α,Ri) = 2α π3/4 exp(−α(r−Ri)2)(3.1) Figure 3.1: Geometry of the distributed gaussian functions according to g number. All distances between closest neighbours is given by the parameter δ. 28 quasi-two dimensional systems Using this setting, all basis functions had the same exponent α and since the set {Ri}M gives rise to a hexagonal pattern the distance between closest neighbours is the same for any two of them. The overlap between close-neighbouring functions is given as S(α , δ) = exp(−αδ2/ 2 ) where δ is the distance between the stated neighbours. One may also rewrite the last overlap function in terms of a dimensionless parameter given as ξ=αδ2 as S(ξ) = exp(−ξ/ 2 ) ; such dimensionless parameter characterizes the resolution of the basis set and controls the value of the entries of the overlap matrix. Due to the fact that too small values for ξ give rise to linear dependency problems (the overlap matrix becomes singular) while too large values imply that the overlap is not good enough to describe a continuous space, the optimal value of ξ has an upper and lower bound. Previous works have shown that this parameter must belong in a narrow interval around ξ= 1.00; in this work, we have taken several values in the interval ξ∈[0.85, 1.15][130, 138–140]. 3.2.2 Quantum mechanical computations In a first step, we have studied one-body problems using isotropic harmonic confinement potentials. For such systems, the general Hamiltonian is given by equation (3.2) which is separable (one term for each coordinate). Therefore, the ground state energy for the isotropic 2D harmonic oscillator is given as E0=k1/2. H(x,y) = 1 2−∂2 x+kx2−∂2 y+ky2=H(x) + H(y)(3.2) Using the basis described in the previous section, we can rewrite the corresponding one-body Schr¨odinger equation as a generalised eigenvalue problem (3.3) where S is the overlap matrix, T is the kinetic energy matrix, V is the potential energy matrix and E is the diagonal matrix containing the eigenvalues of the Hamiltonian while C contains the eigenvectors in its columns. (T+V)C=ESC (3.3) The corresponding matrix elements are given by equations (3.4) , (3.5) and (3.6) respectively; notice that the elements of the kinetic and potential energy depend on the elements of the overlap, thus they can be defined in the same loop. Since the basis function are three dimensional in nature, by solving the general eigenvalue problem, we obtain quasi-2D solutions. In order to correct the obtained energies, since all gaussian have the same exponent, we may factorize out the transverse component of the wavefunction and subtract the corresponding contribution to the kinetic energy, which in this case equals α/ 2 per electron [138]. By doing so, we obtain the corresponding corrected energies which correspond to the equivalent 2D system. 3.3 results and discussion 29 Sij =exp −α 2d2 ij(3.4) Tij =α 2(3−αd2 ij )Sij (3.5) Vij =k 8α(2+α[(x2 i+x2 j) + (y2 i+y2 j)])Sij (3.6) As it has been stated in the previous section, if we set the value of the exponent α=ξ/δ2 for various values of ξ , by means of the variational principle, (given (ξ , g , k) ) we may optimize the ground state energy with respect to the distance between the basis functions δ. 3.3 RESULTS AND DISCUSSION 3.3.1 One particle systems In a first step, we aim to gain some knowledge of how to build an appropriate basis grid exposed in previous section for a given system. That is, for a given potential characterized by the confinement parameter k , a given real number ξ which controls the size of the entries in the overlap matrix and a given number of basis functions represented by g , what is the optimal distance between neighboring basis δopt for describing our system? In order to answer this question, we have taken the following steps: we optimize the one particle ground state energy with respect to distance between neighbouring basis by solving the matrix Schr¨odinger equation for several values of ( k , ξ , g ). We aimed to gain experience on several orders of magnitude for k∈[ 1 × 10 −10 , 9 ] and have employed ξ∈[ 0.85, 1.15 ] with g∈[ 4, 10 ] numbers. A total number of 1190 optimized basis sets were obtained using the Brent optimization scheme for convex functions. As a result of those optimizations, we extracted the first ten eigenvalues of the Hamiltonian and have compared them to the analytical values such that we obtain the relative error given as ϵ=Ei−E0,i E0,i where Ei is the energy of the i -th state and E0,i is the corresponding analytical value. Results for such errors for optimized basis sets are given in figure 3.2. As it can be seen, even if this error gets larger for higher energy states, the maximal error in all the samples is of the order of 5 %. Another feature to be considered is the fact that not only the relative error must be taken into account, but also the fact that analytically degenerate states should remain degenerate in the numerical approach. In order to measure the difference of energy in each energy level for several states given ( k , ξ , g ) we have computed the relative standard deviation of each set and then performed statistics among all standard deviations. As it is shown in figure 3.2, the maximal relative standard deviation Chapter 4 SPHERICAL HOOKE ATOMS Singlet and triplet spin state energies for three-dimensional Hooke atoms, i . e .electrons in a quadratic confinement, with even number of electrons (2, 4, 6, 8, 10) is discussed using Full-CI and CASSCF type wavefunctions with a variety of basis sets and considering perturbative corrections up to second order. The effect of the screening of the electron-electron interaction is also discussed by using a Yukawa-type potential with different values of the Yukawa screening parameter ( λee =0.2, 0.4, 0.6, 0.8, 1.0). Our results show that the singlet state is the ground state for 2 and 8 electron Hooke atoms, whereas the triplet is the ground spin state for 4, 6 and 10 electron systems. This suggests the following Aufbau structure 1 s < 1 p < 1 d with singlet ground spin states for systems in which the generation of the triplet implies an inter-shell one electron promotion, and triplet ground states in cases when there is a partial filling of electrons of a given shell. It is also observed that the screening of electron-electron interactions has a sizable quantitative effect on the relative energies of both spin states, specially in the case of 2 and 8 electron systems, favouring the singlet state over the triplet. However, the screening of the electron-electron interaction does not provoke a change in the nature of the ground spin state of these systems. By analyzing the different components of the energy, we have gained a deeper understanding of the effects of the kinetic, confinement and electron-electron interaction components of the energy. Submitted to IJQC 37 38 spherical hooke atoms 4.1 INTRODUCTION Quantum dots have attracted considerable attention in the last years. The possibility of creating artificial atoms in which the electrons are confined to a center through a quadratic type potential opens the possibility of designing new nanoelectronic devices with properties at will, by precisely controlling the degree of confinement. For instance, transitions never observed in natural atoms can be obtained in the artificial ones, which could be of paramount importance in designing new lasers. [178] Another property that has attracted considerable attention is the determination of the triplet-singlet gap in confined systems, for their use as states of a qubit, or to implement logical gates in quantum computing. The excitation spectrum of two-electron two-dimensional (2D) quantum dots has been investigated by tunneling spectroscopy, [179] and the theoretical prediction of triplet-singlet transitions with increasing magnetic field has been experimentally corroborated. Although less studied, 3D quantum dots are also a subject of interest[86]. Examples of this are magnetically trapped fermion vapors confined by parabolic potentials[180, 181] or quantum defects in diamond crystals used as basic gadgets in quantum computing[182, 183]. One of the simplest and most adequate models used in theoretical studies concerning QDs are the so called Harmonium or Hooke’s atom in which electrons are confined in a spherical harmonic potential [76]. Such models contain parameters that may be tuned in order to represent features corresponding to real QDs [77, 78]. For instance, work carried out in our group using a Hookean exact three-body model to examine electron correlation in a two-electron spherical quantum dot confirmed that triplet-singlet transitions take place as the externally applied magnetic field increases.[79] However, the limitation of using an exact model restricted our study to two-electron systems. That is, the analytical solutions[80] for specific curvature parameter of the two-electron Hooke atoms ( ω2=1 4 , 1 100 . . . ) are well known, which can lead in principle to highly accurate densities for systems in low-correlation regimes ( ω2→ ∞ ) as well as in high-correlation regimes ( ω2→ 0) [35–38]. If similar systems containing larger number of electrons ( Ne> 2) are to be considered, a richer variety of electronic states consisting on several ground state spin multiplicities and non-dynamical electron correlation (multi-determinantal features) arise [39–41]. Although such systems can be employed to understand many-body interactions, the computational cost increases with the size of the system. As in Hooke model atoms, the incorporation of electron correlation effects has been shown to be essential for an adequate interpretation of the experimental spectra and transport properties in Quantum Dots. [81, 82, 84–86] In quantum dots, as opposed to real atoms, the effect of electron correlation may be varied at will through manipulation of the dimension and shape of the nanocrystal as well as of the strength, boundaries and symmetries of the confining fields[87]. Besides, the electron-electron interaction can be screened due to lattice, the doping or the charges induced on the metal gates[89]. This fact makes the quantum dot many-body problem more complex than the more familiar atomic case. 4.1 introduction 39 Finally, Hooke model systems have been repeatedly used in the calibration of electronic structure methods, for they provide very variable dynamic and nondynamic electron correlation regimes [42–44] that pose a great challenge for current computational methods. [37, 45–57] Such calibration has been possible because of the recent availability of highly accurate analytical and benchmark data.[39, 40, 58–64] Several well established methods for the elucidation of atomic/molecular electronic structure have been applied to quantum dots. Salient among these are diagonalizations of large configuration interaction representations of the Hamiltonian matrix (usually referred as “exact”diagonalizations)[184–189], Hartree-Fock (HF)[187, 190–192], coupled-cluster[193], density functional theory[194–197], and quantum Monte Carlo calculations[198]. Let us emphasize, however, that even for two-electron quantum dots,[199] it has been observed that in order to account properly for the electron correlation effects, one must go beyond perturbative schemes based on the independent-particle model or local spin-density functional theory.[179] Last but not least, some of us have found that the use of flexible basis set is crucial for a correct description of strong correlation effects in harmonium.[40, 54, 199] Work carried out in our group using a Hookean exact three-body model to examine electron correlation in a two-electron spherical quantum dot confirmed that triplet-singlet transitions take place as the externally applied magnetic field increases [79]. In the present paper, we address the orbital occupation pattern to establish the Aufbau principle for three-dimensional quantum dots with an even number of electrons ( Ne =2, 4, 6, 8, 10). Full configuration interaction (Full-CI) for Ne =2, and complete active space self consistent field (CASSCF) type wavefunctions, for Ne> 2, are employed to account for electron correlation effects. On top of this, second-order perturbation corrections to these energies are also considered. Both the singlet and the triplet states will be evaluated in order to ascertain whether the ground state wavefunction is either spin unpolarized or spin polarized. This will provide, by the same token, the estimate of the triplet-singlet energy gap as the number of electrons of the quantum dots increases. First, we have employed the Dunning’s family of correlation consistent (CC) basis sets up to sextuple zeta. One should notice that these basis sets are optimized for Coulombic systems, and, therefore, for the sake of consistency, we have compared their accuracy with the available benchmark data. [39, 41, 62, 199] We report the full data as supplementary material. In summary, although these absolute energies do not reach full accuracy, the singlet-triplet gaps were in full agreement with the benchmark data, except for the ten-electron system for which qualitative differences were observed depending on which CC basis set was used. In order to further improve the performance of the basis set, we have optimized a set of even-tempered basis sets (ETBS hereafter) for the Hooke potential considering different number of electrons. After careful inspection, the best balance between accuracy and performance was obtained for the basis set optimized with six electrons in the singlet state, we call this basis set ETBS-6S, and it is the one mainly used throughout the paper. 40 spherical hooke atoms Finally, we also introduce a model to take into account the screening of electronelectron interactions through an electron-electron Yukawa type potential (also known as Debye-Yukawa potential as reference to the Debye-H¨uckel theory so employed in the study of electrolytes and plasmas), as in prrevious works [200]. The screening in electron-electron interaction is often included by introducing an effective dielectric media [201–203]. In the present work, we analyze the use of a Yukawa potential, which at short range is similar to the Coulomb potential, to analyze the effect of the electron-electron screening in correlation effects. We observe that even for an extreme screening, correlation effects are still important, according to the corresponding Coulomb holes, highlighting the importance to adequately treat electron correlation in these systems. 4.2 COMPUTATIONAL METHODS Let us consider the following generalized Hamiltonian operator (in atomic units) for our Ne-electron system: b H=− Ne X i 1 2∇2 ri+ Ne X i 1 2ω2r2 i+ Ne X j>i e−λeerij rij (4.1) where r i is the distance vector between the ith electron and the center of the harmonic potential, which for all the calculations of this paper is centered at the origin. This Hamiltonian represents a harmonically confined Ne -electron system, with a confinement strength ω2 , whose inter-electronic interaction has been screened statically by a Yukawa-like attenuated interaction potential, having a screening length λ−1 ee . Recall that for Ne =2 and λee =0, the Hamiltonian operator of Eq. 4.1 corresponds to the two-electron Hooke atom, which can be separated into its intracular coordinates, namely, the electron-electron relative distance vector r=r 1− r 2 and the center of mass coordinate vector R=1 2(r1+r2) as[80, 204] b H=−∇2 R 4+ω2R2−∇2 r+1 4ω2r2+1 r(4.2) being r=|r| and R=|R| , respectively. Eq. (2) unveils that the center of mass of the electrons will behave as a harmonic oscillator with a spring constant of 2 ω2 and a ground state energy of ER= 3 √ω2 . Likewise, Eq. (2) indicates as well, that the electrons will remain in the proximity of each other for they are retained within finite inter-electronic distances by the potential V(r) = 1 4ω2r2+1 r, (4.3) 4.2 computational methods 41 which is best seen as an effective confinement potential. This model system is commonly known as Hooke atom, Hookean or harmonium.[205] In summary, two types of systems have been considered in this paper: • Coulombic Hooke Atom: The Hamiltonian includes an harmonic confinement term (ω2=0.25) with Coulombic like electron-electron repulsion (λee =0.0). • Yukawa Hooke Atom: The Hamiltonian includes an harmonic confinement term ( ω2=1 4 ) and a Yukawa-type screened electron repulsion ( λee = 0.2, 0.4, 0.6, 0.8, 1.0). The corresponding one-electron confinement integrals and the two-electron Yukawa-type integrals have been implemented by our group in an in-house code and the corresponding integral package interfaced with the GAMESS-US program [206, 207] to perform the calculations described in this work. HF, Full-CI, CASSCF and multireference second-order M¨oller-Plesset (MRMP2) methods were used along with various basis sets of aug-cc type, and optimized even-tempered basis set (ETBS) for 2, 4, 6 and 8 electrons systems. For each of the systems, the HF energy of the singlet state was calculated. The corresponding orbitals were used to perform Full-CI (in the case of 2-electron systems) and CASSCF and MRMP2 calculations in the case of 2, 4, 6, 8 and 10 electrons) for the singlet and triplet spin states. As said in the introduction, we have employed two types of basis sets: i) standard Dunning’s family of correlation consistent (cc) basis sets up to sextuple zeta and ii) even-tempered basis sets (ETBS) optimized for the Hooke atom. The drawback of aug-cc-pVNZ basis sets is that they are optimized for Coulombic systems. Therefore, we have compared their accuracy with the available benchmark data [39, 41, 62, 199] for Hookean systems. We report the full data as supplementary material, but in summary, the conclusion is that, although absolute energies with aug-cc-pVNZ basis sets do not reach full accuracy, the singlet-triplet gaps are in full agreement with the benchmark data. The only exception to this rule is the ten-electron system, which shows a more erratic behaviour with important qualitative differences concernig the ground state spin multiplicity among the various aug-cc-pVNZ basis sets. In order to improve the absolute energies, we have optimized an even-tempered basis set in the presence of a harmonic potential ( ω2= 0.25) considering different number of electrons. We have employed uncontracted ETBS with angular momentum L= 0 to L= 3 and the same number N of primitives per shell. The L and N dependent exponents of the primitives are even-tempered following the scheme: ζk LN (ω2) = ω2 2αL,N(ω2)βL,N(ω2)k−1, 1 ≤k≤N, (4.4) where the parameters α and β are optimized by minimizing the CASSCF (full electron and 13 active orbitals) energies. Similar strategies have been followed in previous publications. [56, 199, 208] 42 spherical hooke atoms After careful inspection, the best balance between accuracy and performance was obtained for the basis set optimized with six electrons in the singlet state, we call to this basis set the ETBS-6S. This basis set can be characterized as an uncontracted 4(SPDF) basis set with the following exponent values: 0.2404032, 0.3130329, 0.4076051, 0.5307491. The total number of basis sets is 80, out of which 68 are linearly independent and these are the only one used in the calculations. The results for this basis set showed a better agreement in both absolute energies and singlet-triplet gaps with respect to the available benchmark data than the standard aug-cc-pVNZ series, and therefore, we focus our discussion on the results obtained with this basis set. 4.3 RESULTS AND DISCUSSION This section is organized in accordance to the number of electrons of the system. Thus, we start our discussion with two electron systems, and then 4, 6, 8, and 10 electron systems follow. In the case of the two-electron Coulombic Hooke atom, the exact energy is known, [80, 204] and therefore, there is a benchmark value to calibrate the basis sets employed throughout this work. In the other cases, we will use the benchmark data available in the literature.[39, 41, 62, 199] Thus, the first section is dedicated to this calibration. Then, the discussion is centered on the triplet-singlet gap for each of the systems, focusing our discussion on the factors that affect this gap, such as the electron repulsion screening and the number of electrons in the system. To understand these trends, we analyze the corresponding natural orbitals, so that we can relate the solution to a specific electronic configuration. Based on this information, we come up with an Aufbau structure to describe the electron filling pattern in these confined systems. 4.3.1 Two-electron systems The results for the two-electron Hooke-type atom with Coulombic electron repulsion are shown in table 4.1. We emphasize that for this system the exact energy for the singlet state is known, [80, 204] namely 2.0 a.u., and, therefore, we use this reference value to calibrate the accuracy of the various basis sets used throughout this work (table 4.1). There is a substantial difference between aug-cc-pVDZ and the rest of the basis sets. Full-CI energy for the singlet state is 2.055213 a.u. with this basis set. The use of aug-cc-pVTZ (25 basis functions including d orbitals) leads to a reduction of the energy error of one order of magnitude, leading to a Full-CI energy of 2.004107 a.u. To further reduce the energy error by one order of magnitude, one has to go up to the aug-cc-pV5Z basis set (2.000476 a.u.) with 105 basis functions. At this point, it is worth noticing that our results are of comparable accuracy to the best unextrapolated result reported by Matito et al. [199] for ω2 =0.25 , namely, 2.0002965 a.u., obtained using systematic sequences of Gaussian primitives with even-tempered 4.3 results and discussion 43 exponents. Finally, the use of the aug-cc-pV6Z basis set, which contains 182 basis functions, leads only to a minor improvement in the energy, 2.000196 a.u. Moreover, this basis set shows large linear dependencies, and in fact, the total number of molecular orbitals in the variational space is reduced to 139 upon elimination of linear dependencies. This fact has led us not to consider this basis set any further in this current work. Besides, two more basis sets have been tested, which will be referred as aug-cc-pV5Z* and aug-cc-pV6Z*. This basis set are created by removing the basis functions with angular momentum greater or equal to four. The reduction in size of the basis set is substantial, from 105 to 75 for aug-cc-pV5Z, and from 182 to 95 for aug-cc-pV6Z. Moreover, the latter basis set upon removal of g and h basis functions does not show linear dependencies. The reduction of basis set size has only minor effects in the energy. In the case of aug-cc-pV5Z basis set, the energy increases only from 2.000476 a.u. to 2.000685 a.u., whereas for aug-cc-pV6Z, from 2.000196 to 2.000426 a.u. Considering the performance of our optimized ETBS-6S basis set, it gives a value for the singlet of 2.000396 a.u., that is, the second lowest energy value and only improved by the considerably higher aug-cc-pV6Z basis set. The triplet-singlet gap ( ∆T−S ), in eV, for two-electron systems can be found in Table 4.1. Irrespective of the basis set employed in the calculation, the lowest energy orbital is of s -type, followed by a shell of p orbitals .[80] We will call to these orbitals the 1 s and 1 p orbitals. Hence, the singlet state is formed by the double occupation of the 1 s orbital and the triplet state corresponds to the promotion of one of the 1 s electrons to one 1 p orbital plus a one-spin flip. The triplet-singlet gap is large. At the Full-CI/aug-cc-pV6Z level of theory, the gap is 9.86 eV and 9.78 eV for Full-CI/ETBS-6S, a very similar value to our reference value of 9.79 eV, which fully justifies the use of our ETBS-6S basis set as a good compromise between accuracy and computational cost. On the other hand, the better performance of the ETBS-6S basis set over the Dunning ones is more evident when more electrons are considered (see below), and therefore, we will discuss in the manuscript the results for the ETBS-6S basis set, in comparison with some values for the aug-cc-pV6Z* basis set. The results for the rest of the Dunning’s basis sets can be found in Table 4.1 and 4.9. Since the use of Full-CI is prohibitive as the number of electrons increases,we have analyzed the performance of CASSCF and MRMP2 methods, using the ETBS-6S basis sets. In Table 4.2, we report the energies obtained at the CASSCF and MRMP2 levels of theory using different active spaces. The results can also be visualized in Fig. 4.1, where we have analyzed the convergence of CASSCF (dashed line) and MRMP2 (continuum line) as we consider more orbitals ( Norb in the active window). We consider from a minimal window of 2 orbitals up to 13 orbitals, which correspond to the 1 s , 1 p , 1 d , 2 s and 2 p shells. The convergence in ∆T−S is obtained quite fast, specially for MRMP2 method, with an excellent agreement with Full-CI results. 44 spherical hooke atoms 4.3.2 Four-electron systems The results for the four-electron systems can be found in Tables 4.3 and 4.9, and in Fig. 4.1. In this case, the use of Full-CI was computationally prohibitive. On the other hand, multideterminantal wave functions are mandatory due to substantial near-degeneracy effects. Consequently, we have decided to use multiconfigurational wave functions of the CASSCF type. We have investigated active spaces that span from 4 to 13 orbitals, corresponding to the 1 s , 1 p , 1 d , 2 s and 2 p shells. We have included the four electrons in this active orbital space and consider all possible excitations within the window that are compatible with the desired spin state (singlet or triplet). The starting orbitals for the CASSCF calculations correspond to the HF orbitals of the singlet state. The biggest calculation using 4 electrons and 13 orbitals for the CAS window leads to 2366 CSFs (Configuration State Functions) for the singlet state, and 3003 CSFs for the triplet state. We have also considered secondorder perturbation corrections based on these CASSCF wavefunctions (MRMP2, hereafter). The electronic structure of the singlet state presents double occupation of the 1s orbital and one of the 1p orbitals. The triplet state corresponds mainly to a configuration in which the 1s orbital is doubly occupied and two 1p orbitals have one electron each. Due to the degeneracy of the 1p shell, the resultant singlet and triplet CASSCF wavefunctions show large nondynamical correlation or near-degeneracy effects, and, therefore, the need to use multiple configurations in the wavefunction. As one can see in Tables 4.3 and 4.9, the values of ∆T−S are negative for all basis sets and methods considered, indicating that in the case of 4 electrons the confinement has led to a triplet-spin ground state. Notice from Fig. 4.1 that the convergence of the results with the window size is quite fast. The values of the triplet-singlet gap at the CAS(4,13) and MRMP2(4,13) levels of theory with the aug-cc-pV6Z* basis set are -1.11 eV and -1.08 eV, respectively, in nice agreement with the reference value by Cioslowski et al. [41] of -1.00 eV. The use of the ETBS-6S basis sets slightly improves these results, giving values of -1.06 and -1.04 eV at the CAS(4,13) and MRMP2(4,13) levels of theory. However, we can see that the CAS(4,13)/ETBS-6S energies are significantly closer to the reference values than the CAS(4,13)/aug-cc-pV6Z* ones. For instance, the energy value for the singlet and triplet states are 6.398720 and 6.359758 , respectively, in good agreement with the values of Cioslowski et al.[41] 6.385543 and 6.348830 a.u., respectively. The introduction of second-order perturbation corrections, albeit non-variational, further improves this agreement, giving energies, 6.390085 and 6.352024, that are even closer to the ones by Cioslowski et al.[41] An advantage of the MRMP2 energies (see Figure 4.1) is that they are less dependent on the active space of the CASSCF wavefunction and, thus, we can use a smaller active space without sacrificing the accuracy. The singlet-triplet gap obtained with MRMP2(4,13) is -1.04 eV, in very good agreement with the one by Cioslowski et al., [41] namely, -1.00 eV. 4.3 results and discussion 45 4.3.3 Six-electron systems The results for the six electron systems can be found in Tables 4.4 and 4.9, and in Fig. 4.1. As in the previous case, we use CASSCF type wavefunctions, with different active spaces that go from a minimal 6 orbital window up to 13 orbitals (1 s , 1 p , 1 d , 2 s , and 2 p ), and include all six electrons. Our largest CAS window involves 26026 CSFs for the singlet state, and 39039 for the triplet. The inclusion of two further electrons in the 1 p -shell yields a very similar tripletsinglet gap and trends close to the ones observed for 4-electron systems. Again, we obtain negative values for ∆T−S , indicating that the ground state is a triplet. For instance, the value of ∆T−S is -1.03 and -1.0 eV at the CAS(6,13)/ETBS-6S and MRMP2(6,13)/ETBS-6S levels of theory. A glance at Fig. 4.1 clearly demonstrates that the results are very well converged with respect to the window size, specially for the MRMP2 level of theory. The values for the aug-cc-pV6Z* basis set are very similar, namely, -1.06 and -1.05 eV. Our results for the singlet-triplet gap agree satisfactorily with the reference calculations for the 6-electron Hookean atom, [62] which yield a value of -0.95 eV with an energy of 12.066294 a.u. for the singlet and 12.031275 a.u. for the triplet. In the case of the six-electron system, ETBS-6S absolute energies again give a significant improvement over the aug-cc-pV6Z* ones. However, now the difference between our CASSCF/ETBS-6S and the reference energies of Strasburger[62] increases with respect to the 4-electron case. For instance, CAS(6,13)/ETBS-6S energies are 12.115772 and 12.077925 a.u. for singlet and triplet states, respectively, whereas the reference energies are 12.066294 and 12.031275 a.u.. However, the introduction of perturbation corrections lowers the energies to 12.082805 and 12.046142 a.u at the MRMP2(6,13)/ETBS-6S level of theory, yielding a tripletsinglet gap only differing by 0.05 eV with respect to the -0.95 eV gap obtained from the data of Strasburger. [62] In summary, although the quality of our absolute energies of each of the states decreases with the increasing number of electrons, the estimation of the triplet-singlet gap remains correct. 4.3.4 Eight-electron systems Tables 4.5 and 4.9, and Fig. 4.1 summarize the results for 8-electron systems, using CASSCF wavefunctions with an active space composed of 8 to 13 orbitals (1 s , 1 p , 1 d , 2 s , and 2 p orbitals) and 8 electrons. In the case of our largest window, the CASSCF(8,13) wavefunction yields 143143 CSFs for the singlet state, and 234234 CSFs for the triplet. The corresponding second-order perturbative corrections can also be found in Table 4.5. In the case of the singlet state, the 8-electron system corresponds to an electronic configuration in which the 1s and 1p shells are fulfilled. In the case of the triplet state, based on the analysis of the occupancies of natural orbitals, an electronic configuration of 1 s2 1 p5 1 d1 -type is observed with ETBS-6S and aug-cc-pV6Z* basis 52 spherical hooke atoms systems, which leads to a reduction of the confinement energy. Our results points to a relationship of the intrashell electron-electron correlation (as it occurs for singlet 4-, 6and 10-electron and triplet 8-electron systems), with the promotion of a more compact electronic cloud. −0.08 −0.06 −0.04 −0.02 0.00 0.02 0.04 0.06 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 N = 2 h(u) u (Å) full, S = 0 full, S = 1 screened, S = 0 screened, S = 1 −0.25 −0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 0.20 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 N = 4 h(u) u (Å) full, S = 0 full, S = 1 screened, S = 0 screened, S = 1 −0.30 −0.25 −0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 N = 6 h(u) u (Å) full, S = 0 full, S = 1 screened, S = 0 screened, S = 1 −0.30 −0.25 −0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 N = 8 h(u) u (Å) full, S = 0 full, S = 1 screened, S = 0 screened, S = 1 −0.30 −0.25 −0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.0 1.0 2.0 3.0 4.0 5.0 N = 10 h(u) u (Å) full, S = 0 full, S = 1 screened, S = 0 screened, S = 1 Figure 4.3: Coulomb holes calculated at the CASSCF( Ne ,13)/ETBS-6S level of theory for both full (solid lines) and λee = 1.0 screened-coulombic (dasshed lines) Hooke systems with different number of electrons ( Ne =2,4,6,8 and 10) and for singlet(black lines) and triplet states (red lines). 4.4 concluding remarks 53 4.4 CONCLUDING REMARKS In the present chapter, we have presented a thorough study of 2-,4-,6-,8and 10electron systems confined in a spherical quantum dot in their singlet and triplet spin states, with the aim of determining the triplet-singlet gap for Hooke-type systems. The effect of screening the electron-electron interaction has also been taken into account effectively by the introduction of a Yukawa type potential. Our results show an interesting pattern in the triplet-singlet gap as the number of electrons increases. Thus, singlet state is the ground state for 2 and 8, whereas the triplet state is the ground spin state in 4 and 6 electron systems. The situation for 10-electron system is again a triplet ground state, but with a gap smaller than the one found for 4 and 6 electrons. Our results can be readily rationalized in terms of the following orbital ordering 1 s < 1 p < 1 d , with a decrease in the successive energy gaps. We have also observed that the screening of electron-electron interaction has a sizable effect, not only on the absolute energies for each state, but also on the triplet-singlet gap. The triplet-singlet gap for the 2 electron and 8 electron cases is specially sensible to the screening effect, favouring the singlet state over the triplet. This can be related to an increase of the corresponding gap between shells upon screening. However, the influence of screening in the triplet-singlet gap for 4-, 6and 10-electron cases was much more reduced, favouring again the singlet state but, in no case, this screening produced a switch between spin states. 54 spherical hooke atoms Table 4.1: HF and Full-CI energies (a.u.) of the singlet and triplet spin states for the two-electron Hooke atom (ω2=0.25; λee =0.0) Basis Size Contraction Singlet HF Singlet FCI Triplet FCI ∆T−S(eV) aug-cc-pVDZ 9 (5s2p/3s2p) 2.087904 2.055213 2.386080 9.00 aug-cc-pVTZ 25 (7s3p2d/4s3p2d) 2.038634 2.004107 2.373537 10.05 aug-cc-pVQZ 55 (8s4p3d2f/5s4p3d2f) 2.038443 2.001484 2.383204 10.39 aug-cc-pV5Z* 75 (9s5p4d3f/6s5p4d3f) 2.038408 2.000685 2.373767 10.15 aug-cc-pV6Z* 95 (11s6p5d4f/7s6p5d4f) 2.038423 2.000426 2.362738 9.86 aug-cc-pV5Z 105 (9s5p4d3f2g/6s5p4d3f2g) 2.038400 2.000476 2.373701 10.16 aug-cc-pV6Z 182 (11s6p5d4f3g2h/7s6p5d4f3g2h) 2.038404 2.000196 2.362579 9.86 ETBS-6S 80 (4s4p4d4f/4s4p4d4f) 2.038400 2.000396 2.359673 9.78 4.4 concluding remarks 55 Table 4.2: HF, CAS(2, Norb ) and MRMP2(2, Norb ) energies with the ETBS-6S basis set, in atomic units, for the singlet and triplet spin states o the two-electron Hooke atom. The RHF energy for the singlet state is 2.038400 and 2.038423 a.u. for the ETBS-6S and aug-cc-pV6Z* basis sets, respectively. Basis Norb CAS(Ne,Norb)∆T−SMRMP2(Ne,Norb)∆T−S Singlet Triplet (eV) Singlet Triplet (eV) 2 2.026787 2.364478 9.19 2.006140 2.360420 9.64 3 2.016225 2.364478 9.48 2.003759 2.360428 9.70 4 2.006551 2.364478 9.74 2.001197 2.360601 9.78 5 2.003410 2.362773 9.78 2.000701 2.360358 9.79 6 2.003056 2.361106 9.74 2.000670 2.359828 9.77 7 2.002705 2.361106 9.75 2.000637 2.359819 9.77 8 2.002358 2.361106 9.76 2.000601 2.359820 9.77 9 2.002019 2.361106 9.77 2.000559 2.359821 9.78 10 2.001681 2.361106 9.78 2.000516 2.359828 9.78 11 2.001630 2.359907 9.75 2.000513 2.359694 9.77 12 2.001280 2.359859 9.76 2.000481 2.359687 9.77 13 2.001082 2.359821 9.76 2.000461 2.359688 9.77 FCI 2.000396 2.359673 9.78 aug-cc-pV6Z* 13 2.001203 2.362886 9.84 2.000483 2.362749 9.86 aug-cc-pV6Z* FCI 2.000426 2.362738 9.86 Reference [80] 2.000000 2.359657 9.79 56 spherical hooke atoms Table 4.3: HF, CAS(4, Norb ) and MRMP2(4, Norb ) energies with ETBS-6S basis set, in atomic units, for the singlet and triplet spin states of the four-electron Hooke atom. The RHF energy or the singlet state is 6.505162 and 6.517905 for the EBTS-6S and aug-cc-pV6Z* basis sets, respectively. Basis Norb CAS(4, Norb)∆T−SMRMP2(4,Norb)∆T−S Singlet Triplet (eV) Singlet Triplet (eV) 4 6.466497 6.420487 -1.25 6.393971 6.357443 -0.99 5 6.436891 6.409056 -0.76 6.395878 6.358352 -1.02 6 6.432908 6.395502 -1.02 6.395253 6.357424 -1.03 7 6.412845 6.387924 -0.68 6.391439 6.356035 -0.96 8 6.416955 6.379714 -1.01 6.392298 6.354322 -1.03 9 6.412845 6.371627 -1.12 6.391439 6.353215 -1.04 10 6.410052 6.368950 -1.12 6.391231 6.352928 -1.04 11 6.409228 6.365185 -1.20 6.391078 6.352579 -1.05 12 6.408315 6.361114 -1.28 6.390980 6.352180 -1.06 13 6.398720 6.359758 -1.06 6.390085 6.352024 -1.04 aug-cc-pV6Z* 13 6.412708 6.371770 -1.11 6.403958 6.364288 -1.08 Reference[41] 6.385543 6.348830 -1.00 Table 4.4: HF, CAS(6, Norb ) and MRMP2(6, Norb ) energies with ETBS-6S basis set, in atomic units, for the singlet and triplet spin states of the four-electron Hooke atom. The RHF energy or the singlet state is 12.253446 and 12.287010 for the EBTS-6S and aug-cc-pV6Z* basis sets, respectively. Basis Norb CAS(6, Norb)∆T−SMRMP2(6,Norb)∆T−S Singlet Triplet (eV) Singlet Triplet (eV) 6 12.177056 12.154962 -0.60 12.091183 12.055694 -0.97 7 12.171843 12.132887 -1.06 12.089361 12.051390 -1.03 8 12.147118 12.117039 -0.82 12.086838 12.050945 -0.98 9 12.137934 12.099985 -1.03 12.085660 12.048808 -1.00 10 12.132401 12.094292 -1.04 12.084591 12.047936 -1.00 11 12.126035 12.087898 -1.04 12.083819 12.047183 -1.00 12 12.120721 12.082893 -1.03 12.083437 12.046636 -1.00 13 12.114932 12.077925 -1.01 12.082804 12.046141 -1.00 aug-cc-pV6Z* 13 12.149632 12.110496 -1.06 12.120797 12.082391 -1.05 Reference[62] 12.066294 12.031275 -0.95 4.4 concluding remarks 57 Table 4.5: HF, CAS(8, Norb ) and MRMP2(8, Norb ) energies with ETBS-6S basis set, in atomic units, for the singlet and triplet spin states of the four-electron Hooke atom. The RHF energy or the singlet state is 19.190980 and 19.247016 for the EBTS-6S and aug-cc-pV6Z* basis sets, respectively. Basis Norb CAS(8, Norb)∆T−SMRMP2(8,Norb)∆T−S Singlet Triplet (eV) Singlet Triplet (eV) 8 19.113340 19.395160 7.67 19.008248 19.272166 7.18 9 19.083985 19.371303 7.82 19.003255 19.270391 7.27 10 19.075987 19.362461 7.79 19.000736 19.268463 7.28 11 19.069697 19.353362 7.72 19.000079 19.269118 7.32 12 19.063989 19.343947 7.62 18.999332 19.269118 7.34 13 19.053174 19.334793 7.66 18.997973 19.268892 7.37 aug-cc-pV6Z* 13 19.110238 19.437774 8.91 19.065563 19.386165 8.72 Reference[39] 19.038 19.430 10.6 Table 4.6: HF, CAS(10, Norb ) and MRMP2(10, Norb ) energies with ETBS-6S basis set, in atomic units, for the singlet and triplet spin states of the four-electron Hooke atom. The RHF energy or the singlet state is 27.932821 and 28.174226 for the EBTS-6S and aug-cc-pV6Z* basis sets, respectively. Basis Norb CAS(10, Norb)∆T−SMRMP2(10,Norb)∆T−S Singlet Triplet (eV) Singlet Triplet (eV) 10 27.828015 27.807466 -0.56 27.675164 27.665330 -0.27 11 27.813009 27.790373 -0.62 27.678880 27.664692 -0.39 12 27.795515 27.773761 -0.59 27.688082 27.663346 -0.67 13 27.777241 27.757081 -0.55 27.689185 27.661990 -0.74 aug-cc-pV6Z* 13 28.020989 28.007019 -0.38 27.951539 27.937691 -0.38 58 spherical hooke atoms Table 4.7: HF, CAS( Ne ,10) and MRMP2( Ne ,10) energies, in atomic units, for the singlet and triplet spin states of the s-Hooke model at different values of the λ screening parameter. λHF CAS(Ne, 10) ∆T−SMRMP2(Ne,10) ∆T−S Singlet Singlet Triplet (eV) Singlet Triplet (eV) 2-electron 0.0 2.038400 2.001681 2.361106 9.78 2.000516 2.359828 9.78 0.2 1.879433 1.845903 2.212892 9.99 1.844644 2.211584 9.98 0.4 1.777898 1.750206 2.131426 10.37 1.748859 2.130299 10.38 0.6 1.710108 1.688101 2.084166 10.78 1.686715 2.083500 10.80 0.8 1.663091 1.645841 2.055639 11.15 1.644470 2.055161 11.17 1.0 1.629427 1.615941 2.037805 11.48 1.614631 2.037480 11.51 4-electron 0.0 6.505162 6.410052 6.368950 -1.12 6.391231 6.352928 -1.04 0.2 4.447821 5.522013 5.481478 -1.10 5.502540 5.464743 -1.03 0.4 5.098802 5.027175 4.988257 -1.06 5.008695 4.972340 -0.99 0.6 4.786937 4.729994 4.692890 -1.01 4.712556 4.678623 -0.92 0.8 4.584830 4.539335 4.505120 -0.93 4.524068 4.492776 -0.85 1.0 4.447821 4.411219 4.380074 -0.85 4.398160 4.369611 -0.78 6-electron 0.0 12.253446 12.133679 12.094292 -1.07 12.084454 12.047936 -0.99 0.2 10.066994 9.960398 9.921242 -1.07 9.909188 9.873006 -0.98 0.4 8.877716 8.792459 8.754633 -1.03 8.745296 8.710196 -0.96 0.6 8.171402 8.104086 8.069290 -0.95 8.064118 8.030753 -0.91 0.8 7.723094 7.671069 7.638238 -0.89 7.636670 7.605815 -0.84 1.0 7.424057 7.382940 7.353003 -0.81 7.354441 7.326141 -0.77 8-electron 0.0 19.190980 19.075987 19.362461 7.79 19.000736 19.268463 7.28 0.2 15.182763 15.084400 15.384014 8.15 15.005631 15.289623 7.73 0.4 13.068595 12.994991 13.317898 8.79 12.922972 13.235696 8.51 0.6 11.834935 11.781507 12.128228 9.43 11.719577 12.060148 9.27 0.8 11.060156 11.021423 11.389781 10.02 10.969653 11.334431 9.93 1.0 10.547028 10.518701 10.905855 10.53 10.475994 10.861138 10.48 10-electron 0.0 27.932821 27.828015 27.807466 -0.56 27.675164 27.665330 -0.27 0.2 21.621750 21.528857 21.511339 -0.48 21.375745 21.363283 -0.34 0.4 18.408818 18.334295 18.317410 -0.46 18.203431 18.190047 -0.36 0.6 16.573893 16.515099 16.499172 -0.43 16.408780 16.395457 -0.36 0.8 15.437499 15.390866 15.376062 -0.40 15.305881 15.293178 -0.35 1.0 14.692340 14.654914 14.641306 -0.37 14.587232 14.575380 -0.32 4.4 concluding remarks 59 Table 4.8: Decomposition of the total energy (a.u.) of the singlet and triplet states into different contributions, kinetic energy, confinement energy (that is the one corresponding to the monoelectronic harmonium confinement operator), and electron-electron repulsion energy. Decomposition of the triplet-singlet gap ∆T−S into kinetic ( ∆K T−S ), confinement ( ∆Vw T−S ) and electron-electron repulsion terms ( ∆Vee T−S ). Two cases are considered: i) Hooke atom with standard Coulombic interactions between electrons and ii) s-Hooke in which the electron-electron interactions are screened by the Yukawa potential with λee =1.0. ∆s−u corresponds to the differences between the screened and unscreened calculation. NeModel Singlet Energy Components (a.u.) Triplet Energy Components (a.u.) Triplet-Singlet Gap (eV) Kin. E. Vconf Vee Total E. Kin. E. Vconf Vee Total E. ∆K T−S∆Vw T−S∆Vee T−S∆T−S 2 Hooke 0.664231 0.888647 0.448803 2.001681 0.920322 1.093808 0.346977 2.361106 6.97 5.58 -2.77 9.78 s-Hooke 0.702495 0.815055 0.098391 1.615941 0.974102 1.027525 0.036178 2.037805 7.39 5.78 -1.69 11.48 ∆s−u0.038264 -0.073592 -0.350412 -0.385740 0.053780 -0.066283 -0.310799 -0.323301 0.42 0.20 1.08 1.70 %∆s−u5.8 -8.3 -78.1 -19.3 5.8 -6.06 -89.6 -13.7 4 Hooke 1.575643 2.661960 2.172449 6.410052 1.575087 2.648067 2.145797 6.368950 -0.02 -0.38 -0.73 -1.12 s-Hooke 1.802488 2.247689 0.361041 4.411219 1.810012 2.234690 0.335372 4.380074 0.20 -0.35 -0.70 -0.85 ∆s−u0.226845 -0.414271 -1.811408 -1.998833 0.234925 -0.413377 -1.810425 -1.988876 0.22 0.03 0.03 0.27 %∆s−u14.4 -15.6 -83.4 -31.2 14.9 -15.61 -84.4 -31.2 6 Hooke 2.298682 4.810967 5.024030 12.133679 2.296848 4.797225 5.000219 12.094292 -0.05 -0.37 -0.65 -1.07 s-Hooke 2.815149 3.794034 0.773756 7.382940 2.821571 3.781367 0.750065 7.353003 0.17 -0.30 -0.64 -0.81 ∆s−u0.516467 -1.016933 -4.250274 -4.750739 0.524723 -1.015858 -4.250154 -4.741289 0.22 0.03 0.01 0.26 %∆s−u22.5 -21.1 -84.6 -39.2 22.8 -21.18 -85.0 -39.2 8 Hooke 2.889288 7.320715 8.865985 19.075987 3.129535 7.489993 8.742933 19.362461 6.54 4.61 -3.35 7.79 s-Hooke 3.755694 5.445591 1.317417 10.518701 4.034318 5.638694 1.232843 10.905855 7.58 5.25 -2.30 10.53 ∆s−u0.866406 -1.875124 -7.548568 -8.557286 0.904783 -1.851299 -7.510090 -8.456606 1.04 0.64 1.05 2.74 %∆s−u30.0 -25.6 -85.1 -44.9 28.9 -24.72 -85.9 -43.7 10 Hooke 3.857152 10.517423 13.453440 27.828015 3.857015 10.511402 13.439049 27.807466 0.00 -0.16 -0.39 -0.56 s-Hooke 5.168257 7.609567 1.877090 14.654914 5.171068 7.603958 1.866280 14.641306 0.08 -0.15 -0.29 -0.37 ∆s−u1.311105 -2.907856 -11.576350 -13.173101 1.314053 -2.907444 -11.572769 -13.166160 0.08 0.01 0.10 0.19 %∆s−u34.0 -27.6 -86.0 -47.3 34.1 -27.66 -86.1 -47.3 60 spherical hooke atoms Table 4.9: Hartree-Fock and CASSCF energies, in atomic units, for the singlet and triplet spin states of the 4, 6 , 8 and 10-electron systems. Calculations done at the CAS-SCF(Ne,13) level of theory. Basis Hartree-Fock CASSCF(Ne,13) ∆T−S Singlet Singlet Triplet (eV) 4 electrons aug-cc-pVTZ 6.508926 6.431129 [1s21p2] 6.378120 [1s21p2] -1.44 aug-cc-pVQZ 6.534156 6.439280 [1s21p2] 6.396548 [1s21p2] -1.16 aug-cc-pV5Z 6.539774 6.436668 [1s21p2] 6.394851 [1s21p2] -1.14 aug-cc-pV5Z* 6.539818 6.436879 [1s21p2] 6.394866 [1s21p2] -1.14 aug-cc-pV6Z* 6.517905 6.412708 [1s21p2] 6.371770 [1s21p2] -1.11 6 electrons aug-cc-pVTZ 12.285452 12.195292 [1s21p4] 12.147453 [1s21p4]-1.30 aug-cc-pVQZ 12.282169 12.167338 [1s21p4] 12.125916 [1s21p4] -1.13 aug-cc-pV5Z 12.315804 12.183503 [1s21p4] 12.144069 [1s21p4] -1.07 aug-cc-pV5Z* 12.315906 12.185233 [1s21p4] 12.145583 [1s21p4] -1.08 aug-cc-pV6Z* 12.287010 12.149632 [1s21p4] 12.110496 [1s21p4] -1.06 8 electrons aug-cc-pVTZ 19.337940 19.271538 [1s21p6] 19.714647 [1s21p52s1] 12.06 aug-cc-pVQZ 19.204578 19.104295 [1s21p6] 19.505586 [1s21p52s1] 10.92 aug-cc-pV5Z 19.261715 19.131805 [1s21p6] 19.493381 [1s21p51d1] 9.84 aug-cc-pV5Z* 19.261851 19.137723 [1s21p6] 19.528024 [1s21p52s1] 10.62 aug-cc-pV6Z* 19.247016 19.110238 [1s21p6] 19.437774 [1s21p51d1] 8.91 10 electrons aug-cc-pVTZ 28.699209 28.615431 [1s21p62s2] 29.258914 [1s21p62s11d1] 17.51 aug-cc-pVQZ 28.186450 28.066966 [1s21p62s2] 28.369080 [1s21p62s11d1] 8.22 aug-cc-pV5Z 28.197306 28.039504 [1s21p62s2] 28.063383 [1s21p62s11d1] 0.65 aug-cc-pV5Z* 28.197450 28.050011 [1s21p62s2] 28.157144 [1s21p62s11d1] 2.92 aug-cc-pV6Z* 28.174226 28.020989 [1s21p61d2] 28.007019 [1s21p61d2] -0.38 * Modified basis set removing basis functions with l≥4 Chapter 5 ML ASSISTED PHASE DIAGRAMS In this chapter we present a systematic procedure to build phase diagrams for chemically relevant properties by the use of a semi-supervised machine learning technique called uncertainty sampling. Concretely, we focus on ground state spin multiplicity and chemical bonding properties. As a first step, we have obtained single-eutecticpoint-containing solid-liquid systems which have been suitable for contrasting the validity of this approach. Once this was settled, on the one hand, we have built magnetic phase diagrams for several Hooke atoms containing few electrons (4 and 6) trapped in spheroidal harmonic potentials. Changing the parameters of the confinement potential such as curvature and anisotropy and interelectronic interaction strength, we have been able to obtain and rationalise magnetic phase transitions flipping the ground state spin multiplicity from singlet (non magnetic) to triplet (magnetic) states. On the other hand, Bader’s analysis is performed upon helium dimers confined by spherical harmonic potentials. Covalency is studied using descriptors as sign for ∆ρ(rC) and H(rC) and the dependency on the degrees of freedom of the system is studied i.e. potential curvature ω2 and inter atomic distance R . As a result, we have observed that there may exist a covalent bond between He atoms for short enough distances and strong enough confinement. This machine learning procedure could, in principle, be applied to the study of other chemically relevant properties involving phase diagrams, saving a lot of computational resources. Published: Xabier Telleria-Allika, Jose M. Mercero, Xabier Lopez, and Jon M. Matxain, ”Building machine learning assisted phase diagrams: Three chemically relevant examples”, AIP Advances 12, 075206 (2022) https://doi.org/10.1063/5.0088784 61 68 ml assisted phase diagrams Figure 5.2: Flowchart for the general procedure for phase diagram construction. 3375 grid points. Finally, in relation to He dimers, broad ranges in interatomic distance R∈[ 1.0, 4.0 ]˚ A and confinement strength ω2∈[ 0.0, 4.0 ] where employed giving rise to a total number of 250000 grid points. Step 2: We select the initialisation points in the grid built in step 1. For the solid-liquid system, a total number of 50 uniform distributed points where selected taking the Cartesian product of 5 equidistant points for x and 10 equidistant points for T . With respect to Hooke atoms, by hypothesis, the edges of the parameter grid would contain all different phases; therefore, since three parameters were employed, the initialisation set was obtained using the extreme 8 points. Meanwhile, for the He dimers, the extrema of the grid were considered along with 100 points selected at random. Step 3: Once a given point in the parameter grid is selected (either the initialisation ones or a new one coming from step 5), we carry out sorting tree or required electronic structure calculations in GAMESS US and we assign a label to this point depending on the result obtained either by the sorting tree, the sign of the energy gap ∆T−S or density laplacian or total energy density. Step 4: We propagate the label information from the data points to the unlabelled ones using 200 trees to form the RF. Step 5: With the information from the previous step, we compute Shannon’s entropy to all guessed points in the grid and select the one with the highest score. We take this point out of the unlabelled points pool U , perform electronic structure calculations as in step 3 and update the labelled database L. 5.3 results and discussion 69 5.3 RESULTS AND DISCUSSION 5.3.1 Solid-liquid phase diagrams with a single eutectic point In a first step, in order to test the validity and usefulness of this approach, we have studied several solid-liquid phase diagrams for systems which include a single eutectic point considering the required idealisations and approximations. Thus, we have employed the methodology presented in work [219] and have chosen some examples involving binary mixtures: Ag/Si, KNO 3 /LiNO 3 and K 2 SO 4 /Li 2 SO 4 . The required data were obtained from the same source [219]. For all cases, as a naive approach, we have computed grids in the (x1 , T) plane taking 100 points for x1 in the [0,1] range and 500 points for T in the [0.25 ×min(Tf,1 , Tf,2) , 1.25 × max(Tf,1 , Tf,2) ] range such that the total number of grid points was 50000. For the initialisation set, we have taken another grid with 5 points for x1 and 10 points for T in the same ranges as the former one; then, we have sampled other 100 points using uncertainty sampling which gives a total number of 150 sampled points. In other words, in order to build a rather accurate phase diagram we have used a 0.3% of points compared to the naive all grid approach. The obtained phase diagrams are represented in figures 5.3, 5.4 and 5.5. Figure 5.3: Ag/Si solid-liquid phase diagram. Green is the heterogeneous solid phase α+β , dark blue is the homogeneous ideal liquid mixture L , light blue is β+L and purple is α+L; the white dots represent the sampled points. As it has been observed in all three studied cases, the uncertainty sampling method samples more thoroughly the region where the larger number of phases coexist, i.e. near the eutectic point. Finding eutectic compositions is crucial for some pharmacological [228–230] 70 ml assisted phase diagrams Figure 5.4: KNO 3 /LiNO 3 solid-liquid phase diagram. Green is the heterogeneous solid phase α+β , dark blue is the homogeneous ideal liquid mixture L , light blue is β+Land purple is α+L; the white dots represent the sampled points. Figure 5.5: K 2 SO 4 /Li 2 SO 4 solid-liquid phase diagram. Green is the heterogeneous solid phase α+β , dark blue is the homogeneous ideal liquid mixture L , light blue is β+Land purple is α+L; the white dots represent the sampled points. 5.3 results and discussion 71 and green chemistry [231, 232] applications. Therefore, this method shall be interesting for discovering novel eutectic formulations. 5.3.2 Ground state spin multiplicity for Hooke atoms For these systems, we have employed a grid in the (λ , ω2 xy , ω2 z) space with 15 points for each dimension in the domain [ 0.05, 2.55 ]×[ 0.20, 0.30 ]×[ 0.20, 0.30 ] , therefore a total number of 3375 grid points. In this case, we have taken just 58 points in the grid to be computed, which is the 1.72 % of all points. We shall start our discussion by describing the obtained ground state spin multiplicity phase diagrams. In figure 5.6 we find magnetic phase diagrams for 4 (left hand side) and 6 (right hand side) electron systems. The vertical axis represents the electron-electron interaction screening parameter λ while the horizontal one represents the ratio between the confinement along the x/y axis and the z axis ε=ω2 x,y ω2 z . We shall proceed to analyse the obtained diagrams. On the one hand, for 4 electron systems, one shall observe that singlets (non magnetic states labelled in red) appear in regions for which ω2 x,y is large, ω2 z is small and, for some (ω2 xy , ω2 z) points close to the boundary, they are more abundant as λ increases. Under the anisotropic potential induced by the fact that ωx,y> ωz the one body energies along the z axis are smaller than in the x , y plane and in virtue of the Aufbau principle, the former will be occupied firstly giving rise to singlet states. Given an anisotropic enough potential for which the ground state is still a triplet (as in the spherical case), we may still obtain a transition to the singlet state by increasing the screening potential λ as by doing so we are turning off the exchange interaction which does not compensate the high spin state and one body interactions will impose singlet spin states. On the other hand, for 6 electron systems it is obvious that some features of the phase diagram differ from the one obtained for the 4 electron system. First, the ground state spin multiplicity behaves in the opposite way as compared to the case for 4 electrons; in this case, singlet states appear for small values for ω2 x,y curvatures and at the same time large values for ω2 z . Besides, the behaviour with respect to the electron interaction screening parameter λ is the same as in the previous case, the higher it is (for a couple of suitable curvatures ( ω2 x,y< ω2 z ) the most likely it is for that potential to give rise to a singlet spin multiplicity for the electronic ground state. In both cases, for large values of the Yukawa-like screening parameter in the electronelectron interaction λ , quantum exchange interaction becomes weaker and both systems show ground state spin multiplicity transition in the close neighbourhood of the vertical line ε= 1. Besides, in the low λ values regime where exchange interaction are not negligible, in both cases we may observe that spin transition does not happen as soon as we distort the spherical symmetry of the potential; we observe there is an inertia to hold the high spin state (triplet S =1 against singlet S =0). This is a natural consequence of exchange interaction as it lowers the total energy of the system via same spin particles; thus the more particles with same spin there are, the more stable the system becomes. Taking it up to the non-interacting electron system (λ→ ∞) , the Hamiltonian is only composed by one-particle operators for which the orbital energy eigenvalues in atomic units are given by 5.8. In this case, depending on the asymmetry parameter ε which is defined 72 ml assisted phase diagrams Figure 5.6: Sampled points in the ( ϵ , λ ) plane. Red indicates singlet ground spin state while blue indicates triplet ground spin state as the ratio between the axial confinement parameters ω2 x,y and ω2 z , the energy of the p orbitals will split as it is shown in Figure 5.7. Taking into account the Pauli’s exclusion principle, we may fit up to a couple of electrons in each orbital and, according to Hund’s rule, in case of degeneracy they will occupy degenerate orbitals such that total spin is maximised. Keeping these two many-body quantum rules in mind, it is easy to see that the ground state spin multiplicity does not only depend on the confinement potential but also in the number of electrons. ϵnx,ny,nz=ωx,y(nx+ny+1) + ωznz+1 2(5.8) When the confinement is sharper along the z direction than in the x and y directions (ω2 z> ω2 x,y) (this is sphericity parameter ε < 1), the degeneracy of the p orbital splits into two main groups, the pz orbital (higher in energy) and the plane composed by the still degenerate px and py (lower in energy). Therefore, when we distort the spherical symmetry in this way nothing essential happens to 4 electron systems as far as ground state spin multiplicity is concerned. However, for six electron systems, as ε gets smaller, the energy of the pz orbital goes higher and we obtain a point in which all px and py orbitals are doubly occupied while pz orbital is empty. If we are to fit all 6 electrons in this potential, we shall see that the ground state spin multiplicity is the singlet; thus, magnetic properties are altered in the process of comprising the sphere. On the contrary, instead of compressing the sphere, if we stretch it along the z axis we obtain a smaller curvature along this direction (ω2 z< ω2 x,y) and the sphericity parameter becomes ε > 1. Once again, according to the one-body energy formula (5.8) this implies the 5.3 results and discussion 73 Figure 5.7: Schematic atomic orbital energy splitting along asymmetry parameter ϵ orbital energies, degenerate in the spherical case, split into two branches: the pz orbital (this time lower in energy) and the plane composed by the still degenerate px and py (higher in energy). Applying the occupancy rules, one shall see that for 4 electron systems the ground state multiplicity is the singlet (all paired electrons) while for 6 electron systems there is no change as long as ground state multiplicity is concerned. As opposed to the previous case, the magnetic properties for the 4 electron system alter along this transformation while the 6 electron system is able to keep its magnetic properties. 5.3.3 Detecting covalent bonding in spherically confined He2 systems In a first step, we have built the covalency phase diagrams shown in Figure 5.8 at HF/augcc-pVTZ level using as label the sign (red for negative and blue for positive) of the laplacian of the density at the bond critical point ∆ρ(rC)and the total energy density at the bond critical point H(rC) . In total, we have computed 500 × 500 grid points in the [ 0.10, 4.00 ]×[ 1.00, 4.00 ] interval in the (ω2 , R) plane taking 104 points for initialisation and 200 more for building the diagram therefore using 0.1216% of all grid points. In these diagrams we represent the confinement curvature ω2 in the vertical axis and the interatomic distance R in the horizontal one. As it can be seen for ∆ρ(rC) , negative values are obtained in regions where the interatomic distance lays around 1.00 and 1.52 ˚ A and confinement curvatures larger than 2.00. On the other hand, as far as H(rC) is concerned, we have found negative values for all interatomic distances shorter than 1.50 ˚ A . Beyond this threshold, we may observe that for large enough harmonic confinement curvature, the sign of the total energy density at the BCP switches from positive to negative; the curvature required to make this switch happen is larger as we increase the inter atomic distance between the He atoms. At this level of theory, for the range in which there are covalency indicators, our model comes in terms with previous works based on all-atom calculations on similar systems. For example for He2 @ C20H20 where the He-He distance is reported to be 1.265 ˚ A and positive 74 ml assisted phase diagrams Figure 5.8: Sign of total energy density for He 2 system computed at HF/aug-cc-pVTZ level, red is for negative and blue is for positive. value for ∆ρ(rC) [210] which is compatible with our calculations for ω2< 2.00. Also for interatomic distances smaller than 1.60 ˚ A , for He2 @ B12N12 and He2 @ B16N16 systems 1.306 ˚ A and 1.456 ˚ A distances are reported [216]. For these two systems, positive ∆ρ(rC) are reported while total energy density is negative in the former (around 1 × 10 −2 a.u.) and zero in the latter which comes along with our results. For larger distances, in a recent work 1.520 ˚ A and 1.546 ˚ A distances are reported for He2 @ C36 and He2 @ C40 respectively [212]; the reported sign of ∆ρ(rC) and H(rC) are positive (of the order of 5 × 10 −3 atomic units). Finally, He atom couples confined in B40 cages have been reported to be 1.672 − 1.640 ˚ A apart [217] and ∆ρ(rC) and H(rC) both turned out to be positive which is compatible with our results for small ω2values. So far, results concerning the nature of bonding in helium dimer obtained by our harmonic confinement model seem to agree with all atom models for several cages. Bearing in mind that real cages may induce a rather small potential curvature ω2 and the fact that switch in the sign of H(rC) happens somewhere in the interval [ 1.40, 2.60 ]˚ A for internuclear distance, we shall focus our attention on studying this region and include electron-correlation effects in our calculations. Hence, we have performed CASSCF(4,8)/augcc-pVTZ calculations in this region. To do so, we have taken a 20 × 20 grid points in the (ω2 , R) plane with evenly separated points in the ( 0.00, 1.00 )×( 1.40, 2.6 ) domain. Using these points, we have computed level maps as in Figure 5.9. In the three maps, we indicate the interatomic distance in the horizontal axis, the confinement curvature in the vertical axis and we have computed level maps for three Bader descriptors (from left to right): ∆ρ(rC) , H(rC) and −G(rC)/V(rC) which take values in different ranges according to each descriptor labelled in red for low values and in blue for higher ones. As we can see for ∆ρ(rC) , we have found positive values in this domain and it increases as the interatomic 5.3 results and discussion 75 distance is shorter an it is not highly dependent on ω2 specially for rather short interatomic distances. For H(rC) we have found negative values which get smaller (more negative) as we shorten the internuclear distance and increase the confinement curvature which is compatible with prior HF calculations as well as with all-atom calculations in which more negative values for this parameter appear when the size of the confining cage is smaller. Figure 5.9: Covalency indicators for MMPT2(4,8)/aug-cc-pVTZ calculated densities for spherical confinement, from left to right: ∆ρ(rC) , H(rC) and −G(rC)/V(rC) . Red shades indicate low values, white indicates medium and blue shades indicate high values. 76 ml assisted phase diagrams 5.4 CONCLUDING REMARKS In this chapter we have been able to produce computationally affordable magnetic and covalency phase diagrams for some few electron Hooke atoms and helium dimers confined in harmonic potentials respectively by sampling the potential feature space by means of semi-supervised learning technologies. Contrary to the conventional way of explicitly calculating every point in a grid, we have taken several initialisation points and have propagated their information using classifiers based on Random Forest, which, compared to label propagation algorithm, has enabled us to study dense grids. By just calculating the points where information is maximal, we have been able to reduce the number of required computations (in our case the required points for computation were below 2% of all grid points). The validation of the method employing RFs was performed by computing analytical ideal solid-liquid phase diagrams involving a single eutectic point. As it has been shown, this method samples all interphases, specially the neighbourhood of the eutectic point in which all four phases coexist. From previous works on 4 and 6 electron systems confined in spherical harmonic potentials with curvature ω2= 0.25, it is known that the ground state spin multiplicity of these systems is a triplet ( S =1) and the first singlet ( S =0) lays around 1 eV above in energy. If we consider the electrons do not interact among them, the whole system behaves as a system composed by n bodies which still must fulfill basic many-body systems features as: Pauli’s exclusion principle, Aufbau filling principle and Hund’s rule. Under these assumptions, we state that ground state spin multiplicity will be imposed by the symmetry of the confining potential and, by altering the symmetry of it, we may obtain ground state multiplicities either singlets or triplets depending on the potential parameters, the strength of interelectronic interactions and the number of electrons. On the other hand, Bader analysis performed upon confined helium dimers by means of harmonic potential seems to be a rather good model which captures the main features of all atom approaches. As far as sign of density laplacian and total energy density is concerned, all trends have been correctly described by this simple model which shows that effective confinement potentials can be employed to capture the essence of complex confined systems and rather simple models can be employed to describe them. This fact enabled the usage of high theoretical level computations upon these systems. As a first step for many other applications of chemical interest, we conclude these machine learning techniques may be useful for classification and exploration. Chapter 6 GAUSSIAN CONFINEMENT AND CONNECTION TO HOOKE ATOMS In this chapter, we have computed and implemented one-body integrals concerning gaussian confinement potentials over gaussian basis functions. Then, we have set an equivalence between gaussian and Hooke atoms and we have observed that, according to singlet and triplet state energies, both systems are equivalent for large confinement depth for a series of even number of electrons n= 2, 4, 6, 8 and 10. Unlike with harmonic potentials, gaussian confinement potentials are dissociative for small enough depth parameter; this feature is crucial in order to model phenomena such as ionization. In this case, in addition to corresponding Taylor-series expansions, the first diagonal and sub-diagonal Pad´e approximant were also obtained, useful to compute the upper and lower limits for the dissociation depth. Hence, this method introduces new advantages compared to others. Submitted to IJQC 77 84 gaussian confinement and connection to hooke atoms the regression. As far as first anharmonic terms g are considered, they are obtained by taking the slope of the linear regression which -in the worst case scenariohas an error of 4 × 10 −4au . If we take a deeper insight of the g values, we may immediately notice that, for a given spin state either singlet or triplet, does not dramatically change with the size of the active space while it is highly dependent on the number of electrons n . Besides the number of electrons, this anharmonic term also depends on the spin state taking the two electron system as the most notorious one. From the previously exposed theory this behaviour was expected since g represents a sum over electrons of an averaged value of a quartic potential with respect to a many-body normalised wave-function; therefore, g condenses a lot of information about the system: the curvature,the number of electrons and the spin state. In the worst case scenario -the one for CASSCF(8,10)(S)/ETBS-6S calculationsthe regression correlation parameter was R2= 0.9623. However, this is a pretty odd case and the average value for this statistic is R2= 0.9991. It can also be seen that even the Hooke atom energy is comparable to the ones obtained by other methods, the anharmonic contribution g is quite different even if we compare it to the one obtained by including dynamical correlation effects via perturbation methods at the same theory level. We have also observed that as soon as the active space size is augmented, the correlation parameter gets rapidly closer to 1 approaching perfect linear dependency. 6.3.2 Loosely confined two electron systems with screened Coulomb interaction Based on the fact that two electron systems with singlet spin state have at least one occupied bound state, we have been wondering at which point of gaussian potential depth the whole systems dissociates ( EG(Vd 0) = 0). On top of this, we have also considered electron-electron interaction to be screened and in what measure it affects the loosely bound system’s stability. Hence, we have modelled these systems using Hamiltonians as in equation (6.17) where the confinement Gaussian potential has been defined as in the previous section and the electronic Coulomb interaction is replaced by a Yukawa-like potential with exponent λ > 0. We have taken 10 values for λ parameter in the evenly separated range [ 0.10, 1.00 ] and 20 values for V0 also in a evenly separated range [− 1.50, − 0.50 ] at MRMP2(2,13)/ETBS-6S level of theory for singlet states; the results for these calculations can be found in figure 6.1. Points with positive energies in this plot are somehow meaningless since positive energies belong to dissociated systems (scattering states); nevertheless, energies are positive and real since the basis function themselves create the Dirichlet boundaries. At a first glimpse, one shall observe that the dissociation limit depth is smaller as λ is larger (therefore electron-electron interaction is weaker). Let us try to make sense of the obtained results. As in the previous section, we have expanded our Gaussian systems energy now taking an additional term as in equation (6.18) where the anharmonic contributions g1 and g2 are generally taken as positive despite the fact the sign is alternating in the original Taylor-like series. Now, we shall obtain EH , g1 and g2 from data using linear regression (omitting all EG> 0 data) via small 6.3 results and discussion 85 Figure 6.1: Energies for gaussian confinement with two electrons in singlet spin state for several screening parameter λvalues. squares minimisation; from the residues, we may notice that they follow an expected cubic polynomial trend due to the fact that we have trimmed the Taylor-like series at that order. H=−1 2 2 X i=1∇2 i− 2 X i=1 V0e−ω2 2V0r2 i+e−λr12 r12 (6.17) EG+2V0=EH+g1 V0 +g2 V2 0 +Oω8 V2 0(6.18) Once we have obtained the coefficients we can solve the equation (6.18) for EG as in equation (6.19) where we still have a Taylor-like series. Now, we can exploit an interesting property of the energy function: in the Taylor-like series signs are alternating, therefore it is a Stiljies function. That means if we obtain the main Pad´e sequence (i.e. the sequence containing the diagonal and lower diagonal Pad´e approximants PN N1 V0 and PN N+11 V0 respectively), we shall obtain physically relevant values such as the dissociation limit of the system. Since we only have obtained the first three coefficients of the Taylor-like series, the highest order main Pad´e sequence we shall obtain is the one composed by the diagonal approximant P1 1(V−1 0) and the subdiagonal approximant P1 2(V−1 0) given in equations (6.20) and (6.21) respectively. It is known that the sequence P1 2(V−1 0) will converge to the right energy from bellow while P1 1(V−1 0) will converge from above. For obtaining the dissociation limit, we take these Pad´e approximants solve them for V0 obtaining the lower and upper limit of the dissociation depths Vd− 0 and Vd+ 0 respectively in terms of the physical quantities EH , g1 and g2 for a given screening parameter λ . All obtained results are condensed in table 6.2. 86 gaussian confinement and connection to hooke atoms EG V0 =−2+EH V0 +g1 V2 0 +g2 V3 0 +Oω8 V4 0(6.19) P1 11 V0=−2g1+EH−2EHV0 g1−EHV0 (6.20) P1 21 V0=(4g1+2EH)V2 0−(4g2+4g1EH)V0 −(E2 H+2g1)V2 0+ (g1+2g2+EH)V0+ (g2EH−g2 1) (6.21) Table 6.2: Hooke atom energy ( EH ), first anharmonic terms ( g1 , g2 ) and bound dissociation limits obtained for several screening parameter values for 2 electron systems with singlet spin state at CASSCF(2,13)/ETBS-6S and MRMP2(2,13)/ETBS-6S levels. All values are given in atomic units. EHg1g2Vd− 0Vd+ 0 λCASSCF MRMP2 CASSCF MRMP2 CASSCF MRMP2 CASSCF MRMP2 CASSCF MRMP2 0.1 1.9379 1.9372 -0.3839 -0.3839 0.0163 0.0164 0.731 0.730 0.771 0.770 0.2 1.8719 1.8712 -0.3829 -0.3830 0.0200 0.0201 0.689 0.688 0.731 0.731 0.3 1.8204 1.8197 -0.3802 -0.3803 0.0232 0.0233 0.657 0.657 0.701 0.701 0.4 1.7770 1.7753 -0.3712 -0.3691 0.0232 0.0220 0.635 0.634 0.680 0.680 0.5 1.7397 1.7394 -0.3578 -0.3587 0.0203 0.0208 0.618 0.617 0.664 0.663 0.6 1.7143 1.7135 -0.3568 -0.3569 0.0233 0.0234 0.603 0.603 0.649 0.648 0.7 1.6902 1.6890 -0.3487 -0.3482 0.0223 0.0222 0.593 0.592 0.639 0.638 0.8 1.6702 1.6686 -0.3418 -0.3406 0.0215 0.0211 0.584 0.584 0.630 0.630 0.9 1.6532 1.6514 -0.3354 -0.3339 0.0206 0.0201 0.578 0.577 0.624 0.624 1.0 1.6376 1.6369 -0.3280 -0.3281 0.0192 0.0192 0.573 0.572 0.619 0.618 If we focus our attention upon a given λ value and study a given estimated physical property, we shall notice that including dynamic correlation effects via perturbation methods does not quite make a big difference with respect to the same quantity obtained by regular CASSCF method. Now, as far as EH is concerned, this energy is smaller as λ increases which is to be expected for Hooke atoms. In contrast to the stated former results, in this very work, we obtained a Hooke two electron singlet atom energy for λ= 0.2, 0.4, 0.8 and 1.0 to be EH= 1.8459, 1.7502, 1.6881, 1.6458 and 1.6159 a.u. respectively at CASSCF/ETBS-6S while in table 6.2 the obtained energies are in average 0.025 a.u. higher. As we have discussed in the previous section, accurate Hooke energies are obtained for deep potentials, nevertheless, in this section we have been dealing with loosely confined systems. Therefore, on the basis of this approximation, we may state that our estimations are rather reasonable and both regimes have pretty unique features. As for the anharmonic terms, g1 we may obverse it also gets smaller as λ increases. We may hypothesise that as electron-electron interaction gets weaker, correlation effects are also turned off and electrons are more likely found in the centre of the potential well, thus this first anharmonic term becomes smaller. On the other hand, the second anharmonic term g2gets a maximum for λ=0.6 and then decreases. 6.3 results and discussion 87 Figure 6.2: Limit dissociation potentials for several λvalues. Finally, the lower and upper bounds for the limit ionization potentials have in the worst case scenario a 0.046 a.u. amplitude as we may observe that we have predicted their behaviour in terms of physical constants by setting the corresponding Pad´e approximants (6.20) and (6.21) to zero and solving for V0 . We shall see that these limit potentials are shallower and asymptotic to a limit value at which only one-body interactions are relevant. Thus, we get an obvious conclusion, as electron-electron interaction is turned off the potentials does not need to do so much ”work” to confine the interacting particles and shallower potentials are still able to confine them. We may find a visual summary in figure 6.2. 88 gaussian confinement and connection to hooke atoms 6.4 CONCLUDING REMARKS In this work we have computed and implemented the required one-body integrals for quantum particles confined in gaussian potential wells for which the centre of the basis function and the centre of the potentials do not need to coincide. Such implementation has been interfaced to electronic structure software GAMESS-US so that we can make use of its quantum chemical computation machinery to study systems of electrons confined in dissociative potentials. Firstly, we have performed computations on deeply confined systems (large V0 parameter) with controlled width parameter such that the curvature of the potential at the minimum point was ω2= 0.25. Since previous results on harmonic potentials have been well established for n= 2, 4, 6, 8 and 10 electrons, by means of Taylor series we have shown our calculations are compatible with the former ones. Finally, we have studied dissociative systems composed by two electrons in which the conventional Coulomb operator was substituted by Yukawa potentials. In this case, we have not only obtained the corresponding Taylor-series expansion but also the first diagonal and sub-diagonal Pad´e approximant which were useful to compute the upper and lower limits for the dissociation depth for several screening parameters λ. Chapter 7 SUMMARY, MAIN CONCLUSIONS AND FUTURE WORKS In this last chapter, we have not only summarized the main results and conclusions obtained in the present research work, but also have sketched new research lines based on the gained knowledge. In particular, we have focused our attention on the possibility of applying gaussian confinement potentials so as to model endohedrally doped cage clusters. In this case, the doping atom is explicitly represented in the model Hamiltonian while the confinement potential can be adapted so as to mimic properties of the all atom system. Hence, the potential parameters, namely (V0 , β) can be chosen such that relevant properties like ionization energies, electron affinities and singlet-triplet gaps obtained from employing full Hamiltonian approaches can be recovered from the simplified models. With the aim of obtaining accurate results without compromising the computational capacities, the developed machine learning methods can be adapted and extended for building reliable simplified models. On the one hand, optimization schemes similar to the ones described in chapters 3 and 4 can be employed to obtained optimal basis functions given some input potential parameters and doping agent’s atomic number. On the other hand, if some properties such as lowest laying spin state multiplicity, the sign of singlet-triplet gap or the sign of magnetic coupling constants change by varying the external potential parameters, methods presented in chapter 5 can be employed for building phase diagrams for such properties efficiently. 89 90 summary, main conclusions and future works 7.1 MAIN CONCLUSIONS In this thesis work some advances concerning systems of electrons confined by static potentials have been provided. First, computational protocols for studying systems of few electrons with several total spin numbers confined in one, two and three dimensional harmonic potentials have been developed. In particular, on the one hand, adequate either distributed or one-centre even-tempered gaussian basis functions have been obtained by means of classical convex optimization techniques. On the other hand, these basis functions along with the employed multi referential CASSCF and MRPT2 methods have been proven to accurately describe Wigner location phenomenon for weak confinement potentials, ground state spin multiplicities, Aufbau structure and singlet-triplet energy gaps. For spherical Hooke atoms, based on the Aufbau structure of the system, the spin multiplicity of the lowest energy configuration purely depends on the number of electrons. However, if spherical symmetry is broken such that the harmonic confinement along a given axis differs from the confinement along the perperndicular plane (i.e. ω2 z=ω2 x,y ), the ground state spin multiplicity does not only depend on the number of electrons, but also on how spherical symmetry has been broken. Given a number of electrons, a machine learning method based on semi-supervised learning and uncertainty sampling has been develop in order to obtain ground state spin multiplicity phase diagrams efficiently in the space composed by the stated confinement parameters and the electron-electron screening parameter. In addition, this method has also be proven to be performing when applied on further chemically relevant systems such as solid-liquid binary systems and chemical bonding analysis. As an extension to harmonic confinement potentials, gaussian confinement potentials have been introduced so as to properly describe many-centre molecular features, dissociation processes and anharmonic contributions. The required one-body integrals have been implemented in GAMESS US ; the validity of such implementation has been proven by comparing the equivalent Hooke atom energies with the ones obtained in previous works in the literature and our own results. The results and methodologies developed in the present work can be extended to further systems such as ones consisting on different confinement regimes, larger number of electrons with richer number of ground state spin multiplicities and employed the obtained models to more realistic systems by appropriately fitting the potential parameters to all-atom systems. 7.2 further works 91 7.2 FURTHER WORKS In a nutshell, in this thesis we have developed computational protocols which lead to: obtaining optimal basis sets for harmonically confined one-, two-, and three-dimensional systems; implementing machine learning algorithms for optimizing some of the stated basis functions obtaining phase diagrams efficiently and implementing and proofing the validity of gaussian confinement potentials. With the aim of extending the gained knowledge to other systems and based on the expertise of our group, we have considered to apply the developed models on the study of endohedrally doped cage clusters. We shall now introduce the two main topics concerning this work: endohedrally doped cage clusters (so related to artificial atoms or superatoms) and quantum systems confined by static potentials. After discussing some works and results obtained in both fields, we will establish a bridge between both disciplines and present our approach to do so. On the one hand, clusters are somewhere between bulk materials and molecules, which makes them have interesting electronic properties. Among these systems, we may find the spherical hollow nanoclusters which may be doped as it was firstly done for lanthanum in fullerene-60 La@C 60 [270] followed by other similar systems [271, 272] and the so interesting N@C 60 which potentially can be used as an atomic clock [273]. Besides fullerene-basedspecies, we may also find systems called ”superatoms” [274–277] as Al 12 cages which depending on the dopant can give rise to superhalogens (using B as dopant), super alkali metals (using P as dopant), superchalcogen (using Ca as dopant) or stable 40 electron species using Si as dopant [278]. Other interesting systems are the ones based on semiconductor clusters for which confinement-induced properties such as tunable absorption, emission and photoluminescence have been observed [279–285]. In line with superatom systems, ”atomic” properties arise from collective behaviour of the whole electronic structure; nevertheless, some species such as atoms confined in semi-conducting clusters [286–290] for which the cluster acts only as a confinement, atomic properties are very similar to the ones observed by isolated dopant elements. On the other hand, gaussian potential wells have been employed in rather theoretical electronic structure studies [252–256] as well as in more ”applied” condensed matter studies concerning: quantum dots [233–237], thermo-magnetic properties [239–242] and interactions with electric and laser fields [243–248]. These potentials are pretty similar to harmonic potentials so employed in benchmark for electronic structure methods [36, 37, 41, 80, 261–266] with the subtle difference that they have a finite number of bound states and can give rise to molecular structure (chapter 6). Bearing these two concepts in mind, as a first step, we would like to model confined alkali metal and halogen atoms in gaussian potentials and study their behaviour concerning ionization energy and electronic affinity in material clusters. We have obtained approximated estimates for such energies based on free atoms’ ionization energy, their highest occupied atomic orbital and the gaussian confinement parameters by using the Helmann-Feyman theorem as well as first order perturbation methods and checked the validity of such approximations by means of explicit calculations. 92 summary, main conclusions and future works 7.2.1 Some preliminary results concerning endohedral systems The general Hamiltonian for an atom with nuclear charge Z and n electrons confined in a gaussian potential with depth and width parameters V0 and β respectively can be written as in equation (7.1). H=−1 2 n X i=1∇2 i−Z n X i=1 1 ri + n X j>i 1 rij −V0 n X i=1 e−βr2 i(7.1) If we consider monodeterminantal wave functions, apply the adiabatic approximation (Koopmans’ theorem) and use the Helmann-Feynman theorem, we obtain an approximate variation of the ionization potential with respect to confinement depth in terms of the confinement potential paramenters and the nth atomic orbital: ∂I ∂V0 =−*Ψion  n−1 X i=1 e−βr2 iΨion++*Ψatom  n X i=1 e−βr2 iΨatom+ ≈Dϕ(rn)e−βr2 nϕ(rn)E Thus, if we consider V0 as a small parameter, we may integrate the result obtained by Helmann-Feynman theorem in the range V′ 0∈[ 0, V0] such that we obtain an approximate value for the ionization potential I in terms of the unconfined ionization potential I0 , the confinement potential and the nth atomic orbital as in (7.2) . Of course, in this approximation we are taking for granted that the shape of the atomic orbital and, in general, the wave function is uncoupled to the potential parameters V0 and β which is not true according to electronic structure computations concerning Hooke atoms (chapters 4 and 5). Thus, the curvature of the potential defined as ω2= 2 βV0 may be an interesting parameter for future discussions. As an alternative formulation, one shall apply perturbation methods upto first order correction for energy and keep the unperturbed wave function using V0 as the required small parameter. Still in a monodeterminantal formulation and using the adiabatic approximation, it can be shown that one reaches the same result for approximated ionization energies. I≈I0+V0Dϕ(rn)e−βr2 nϕ(rn)E(7.2) Using a similar strategy, if we consider the neutral atom is composed by n electrons and, therefore, the anion by n+ 1 electrons, electron affinity A can be defined as the negative sign of ionization energy of the anionic species and therefore it is straightforward to see that it can approximately be expressed as in equation (7.3). A≈A0−V0 nDϕ(rn+1)e−βr2 n+1ϕ(rn+1)E(7.3) 7.2 further works 93 As it can be seen from both equations (7.2) and (7.3) , considering V0 is small enough, if β is very small i.e. a low curvature confinement, the effect of the potential on the system is such that it shifts the ionization energies by a factor of V0 . On the contrary, if β is large enough, the potential will have no effect on the system since, in the limit, it approaches a null measure index function. Now, the key point is to compute the integral in equations (7.2) and (7.3) in which information about electronic structure and confinement potential is encoded. As it is commonly done in quantum chemistry, we expand basis functions in terms of contracted gaussian primitive functions and by electronic structure computations we obtain the coefficients of such basis orbitals to obtain atomic (molecular) orbitals. Hence, we shall rewrite the required integral as a bilinear form as in (7.4) where the matrix V contains information about the interaction between the pth and qth primitive gaussian functions with the confinement potential (these integrals are given in chapter 6), the matrix C contains the qth primitive expansion coefficients for the ith basis function and the vector Φ contains the expansion coefficients of the basis functions for the nth electron obtained by electronic structure calculations. Dϕ(rn)e−βr2 nϕ(rn)E=X jipq c† jcic(j)† pc(i) qZg† pgqe−βr2 nd3r= (CΦ)†VCΦ(7.4) With the aim of exploring if these approximations are valid at HF/6-311++G level for alkali metals M=Li,Na,K and MP2/6-311++G level for halogens X=F,Cl in order to describe ionization potentials and electron affinities in confined systems. We have also carried out calculations concerning H atom at HF/6-311++G level so that we could gain a better picture concerning ionization potential. In order to obtain a general picture, we have built a 20 × 20 grid in the (V0 , β)∈[− 0.500, − 0.001 ]×[ 0.10, 2.00 ] domain and have computed the error ε in each point by comparing the explicitly obtained ionization energy I with the one obtained by means of formulas (7.2) and (7.3) , namely Itheo as ε=|I−Itheo| |I|× 100. As it can be seen in figure 7.1, the obtained error does not only depend on the confinement potential parameters but also on the atomic number of the atom we are studying. For all cases the obtained error is smaller for small values of the potential depth parameter V0 , which is expected from the perturbation-method scheme. Another shared feature is the fact that, in general, the error gets smaller for larger values of the width parameter β ; we may also highlight the cases concerning Na and K for which the error seems to have a minimum for some values of the width parameter. So far, we have taken the first steps to study this kind of systems. There are many ways in which our predictions can be improved: extend the theory to multi reference wavefunctions, obtain optimized basis sets for each element given the confinement parameters (V0 , β) in a similar way as in chapter 3, compute further terms in the perturbation series and so on. 100 euskarazko laburpena 8.1.3.2 Erdi-gainbegiratutako ikaskuntza Gainbegiraturiko ikaskuntza teknika gehienei esker lorturiko iragarpenen eta sailkapen eredu zehatzak izaten dira, baldin eta ikaskuntza-datuen kopurua nahiko handia bada. Hala eta guztiz ere, errealitatean agertzen diren egoera askotan, (gure kasuan bezala) ez-menpeko aldagaiaren etiketak lortzea denbora aldetik edo ekonomikoki garestia izan daiteke. Beraz, ezagutza eskuragarrian oinarrituriko esperientzia gehiago lortzea eta gure baliabideak (borondatea, denbora eta dirua) gure ezjakintasunaren zokorik ilunenak argitzen inbertitzea zuhurra da. Erdi-gainbegiraturiko eta ikasketa aktiboko teknikak datuen arabera aplika daitezke, eta horretarako bi multzo desberdin ditugu. Erdi-gainberiraturiko ikaskuntza eta ikaskuntza aktiboko teknikak bi datu sorta desberdin ditugun kasuetan aplika ditzakegu: etiketaturiko sorta bat L eta etiketatu gabeko beste sorta bat U . Izenak adierazi bezala, ekiketaturiko data sorta ezugarri eta etiketen biderkadura Cartesiar ordenatuz osaturiko (x1 , y1). . . (xl , yl) itxuradun puntuek osatzen dute, non YL={y1. . . yl}∈{ 1 . . . C} kategoria posible guztien sorta den. Bestalde, etiketa gabeko eta (xl+1 , yl+1). . . (xl+u , yl+u) puntuez osaturiko sorta U ere badugu, non YU={yl+1. . . yl+u} ∈ { 1 . . . C} etiketak ezezagunak diren eta sorta honen kardinalitatea aurrekoarena baino askoz handiagoa den l << u. Horrela, lan oso berrietan egin den moduan [126, 127], fase diagramak erakitzeko asmoz, hurrengo esperimentua zein baldintzapetan egin jakiteko datu sorta hauetan dagoen informazioa erabil genezake. Hasierako hipotesiak kontuan izanda, parametro espazioa diskretizatuz X={x1. . . xl+u} itxurako sare bat eraiki dugu, non lehenengo l puntuak hasierako konputazioen bidez Y={ 1 . . . C} sortan agertzen diren kategoria guztiak agertzen direlarik etiketatuko ditugun eta gainontzeko u puntuak (hasiera batean) etiketatu gabe geratuko diren l << u . Jarraian, lehen l puntuetako etiketen informazioa gainontzeko puntuetara hedatuz, etiketatu gabeko puntuek C kategoria posible guztietan egoteko duten probabilitate banaketak lortuko ditugu, adibidez, etiketen hedapen algoritmoa erabiliz. X. Zhu-k eta Z. Ghahramani-k ”etiketen hedapena” garatu zuten algoritmoari esker [128] etiketatutako datuen informazioa etiketatu gabeko datuetara Markov-en prozesu baten bidez hedatzen da. Hor, etiketatutako eta etiketarik gabeko datu guztiak biltzen dituen grafo bat eta binakako elkarrekintzak modulatzen dituzten pisuak definitzen ditugu. Markoven kate homogeneoen eta matrize estokastikoen propietateak erabiliz, edozein hasierako baldintza Y(0) hartuta ere, algoritmoak soluzio geldikor bakar batera konbergitzen duela frogatu zuten (8.5). Yu= (I−˜ Tuu)−1˜ TulYl(8.5) Metodo horren desabantaila nagusia (I−˜ Tuu)−1 matrizearen konputazioan datza, erabilitako sareak nahiko handiak edo dentsoak badira, objektu horren konputazioa garestia edo ezinezkoa izan daiteke. Arazo hori saihesteko, (8.5) ekuazioan agertzen den formula itxia erabili beharrean, hasierako etiketa bektore bat hartu eta Markov-en ereduan irteratuz informazioa hedatzea lortzen dugu. Hala eta guztiz ere, matrize handiak erabiltzerakoan konbergentzia lortzeko hainbat iterazio egin behar direnez, algoritmo horrek astuna izaten jarraitzen du; batez ere, sareen dentsitatea edota dimentsio kopurua handitzen baditugu. Alternatiba gisa, konputazionalki merkeagoak diren klasifikatzaileak (esaterako, ausazko 8.1 sarrera 101 basoak) problema hori ebazteko tresna aproposak dira. Gain-doitze arazorik izaten ez dutenez, zuhaitz kopuru nahiko handia erabiliz lan honetan ikertutako sistema guztiak ikasteko metodo unibertsalak dira. Gauzak horrela, hasierako datu sorta bat erabiliz ausazko basoetan oinarrituriko modelo bat entrenatzen dugu; hori erabiliz, etiketatu gabeko puntuek eta kategoria bakoitzean egoteko duten probabilitateak eskuratzen ditugu. Horrela eginik, etiketen hedapen algoritmoa erabiltzearen baliokidea den metodo bat ezarri dugu. Ikastea honetan datza: gure ezjakintasuna sakonena den alorretan informazio gehiago lortzea alor horri buruz dugun ezagutza zabaltzeko. Modu eraginkorrean ikasi ahal izateko, norberak ezagutza eza duen puntuetan arreta jarri behar du. Ziurgabetasun laginketa ziurgabetasun handiena duten puntuen laginak hartzea ahalbideratzen digun metodoa da, baina, nola kuantifika genezake ziurgabetasuna? Demagun probabilitate banaketa funtzio bat dugula PC(y|x) zeinek, parametro bektore bat x emanik, { 1 . . . C} kategorietan egoteko y probabilitateak ematen dizkigun. Horrela, eskuratutako propabilitateak erabiliz, puntu jakin baten ziurgabetasuna hainbat eratan defini dezakegu: konfiantza txikiena (8.6) , ziurgabetasun marginala (8.7) eta Shannon-en entropia (8.8) . Lehenak etiketarik gabeko datuen aukerarik txikiena (horregatik zalantzagarriena) maximizatzen du; bigarrena aurrekoa baino malguagoa da, bi iragarpenik probableenak kontuan hartzen baititu. Azkenik, Shannon-en entropia informazio edukiaren neurria da; aldagai batek zenbat eta informazio gehiago izan, honen naturari buruz orduan eta gutxiago dakigu [129]. uLC (x) = 1−max CP(C|x)(8.6) uMS (x) = 1−[P(C1|x)−P(C2|x)](8.7) uSE (x) = −X C P(C|x)log P(C|x)(8.8) 102 euskarazko laburpena 8.2 LAN HONEN HELBURUA Wigner-en molekulei, Hooke-n atomoei eta makinen ikaskuntzako teknika batzuei buruzko artearen egoera deskribatu ondoren, atal honetan oraingo lanaren helburu nagusiak jasoko ditugu: 1. Dimentsio batean eta bitan indar-parametro desberdinen ( k=ω2 ) bidez potentzial harmonikoetan konfinaturik dauden sistemak deskribatzeko erabiltzen diren oinarri funtzio gausstar banatuak optimizatzea ML teknikak (sare neuronalak) erabiliz. Horretaz gain, egitura elektronikoa lortzeko ahalbideratzen duten metodo optimoak eskuratzea eta horiek n={ 2, 3, 4 } elektroidun spin garaiko sistemak ikertzeko erabiltzea. 2. Zentru bateko oinarri funtzioak teknika klasikoen bidez (simplex eta Newton-Raphson) optimizatuz hiru dimentsiotako n={ 2, 4, 6, 8, 10 } elektroidun eta k=ω2= 1 / 4 potentzial parametrodun Hooke-n atomoen deskribapen egokia eskuratzea eta horri dagokion metodo optimoan oinarrituta, oinarrizko singlete eta tripletearen energia zehatzak erdiestea. 3. Makinen ikaskuntza-metodoak hobetzea eta inplementatzea. Erdi-gainbegiratutako ikaskuntzan eta ziurgabetasun laginketa tekniketan oinarrituta, fase diagramak eraginkortasunez lortzea eta kimikoki esanguratsuak diren sistema batzuk hornitzea. 4. Konfinamendu potentzial gausstarrei eta oinarri funtzio gausstarrei dagozkien gorputz bakarreko integral analitikoak kalkulatzea eta GAMESS US kodean inplementatzea. Aurreko ataletan lorturiko emaitzekin lotura duten kalkuluak burutzea. 8.3 egindako lanen laburpenak 103 8.3 EGINDAKO LANEN LABURPENAK 8.3.1 Sasi dimentsio bateko sistemak Aurreko lan batzuetan, potentzial esplizitu gabeko eta elektroi gutxidun sistemak aztertu zituzten kutxa-luzera aldakorrak erabiliz, eta korrelazio elektuak ROHF (geruza irekiko Hatree-Fock uhin funtzio mugatua) eta FCI (konfigurazio-elkarrekintza osozko uhin funtzioa) uhin funtzio metodoen bitartzen ikertu zituzten [130]. Sistema horiek berezko interes teorikoa badute ere, lan honetan ikerturiko elektroi kopuru berberak dituzten sistemak esperimentalki behatu dituzte [131, 132]. Lan honen lehen kapituluan, sasi dimentsio batean potentzial harmonikoen bidez konfinaturik dauden elektroi gutxidun ( n={ 2, 3, 4 } ) sistemen lokalizazio propietateak aztertu ditugu. Horrela, bada, sistema horiek modelizatzeko, (8.9) ekuazioan agertzen den eragile Hamiltondarra erabili dugu. H=−1 2 n X i=1∇2 i+k 2 n X i=1 r2 i+ n X i=1,j>i 1 rij (8.9) Literaturan jakina denez, dimentsio batean Coulomb-en eragilea singularra da [136, 137]; arazo hori konpontzeko, hiru dimentsionalak diren eta lerro baten hedatzen diren s motako oinarri funtzioak erabili ditugu. Hori dela eta, dimentsio puru bat izan beharrean, sistema sasi-unidimentsionala dela diogu. Aurreko lanetan forgatu duten moduan, oinarri horiek erabiliz lorturiko energiari sobera dauden zeharkako osagai bien ekarpena kenduz, sistemaren energia zuzendua lortzen dugu [138]. Kasu honetan, lortutako uhin funtzioak era egokian errepresentatzeko asmoarekin, x= 0 puntuan zentraturiko eta x noranzkoan hedatzen den dimentsio bateko tarte erregulardun sare batean zehar s motako funtzio gausstar normalizatuak kokatu ditugu. Oinarri funtzio horiek (8.10) ekuazioaren bidez adierazita daude; hemen, Rµ= (xµ , 0, 0 )T bektoreak µ funtzioaren zentru-koordenatuak ematen ditu. ϕµ(r;α,Rµ) = 2α π3/4 exp(−α(r−Rµ)2)(8.10) Era honetan definituriko segmentuaren luzera beteriko egoeraren energia garaieneko itzulera puntuak x0(n , k) = ±(2n+1)2 k1/4 erabiliz definitu dugu. Puntu horietariko bakoitzean eta x= 0 puntuan funtzio bana jarri ondoren, tarte erregularretan banaturiko beste 4 m funtzio jarri ditugu eta itzulera puntu klasiko bien artean 2 m daude. Hortaz, guztira M= 3 + 4 m s motako funtzio jarri ditugu eta alboz alboko funtzioen arteko distantzia δ=2x0 m+1 izan da. Funtzio guzti horiek α berretzaile bera dutenez, ondoz ondoko funtzioen arteko gainezarpen integrala S(α , δ) = exp(−αδ2/ 2 ) = exp(−ξ/ 2 ) da. Lan honetan, aurreko hainbatetan oinarrituz [130, 138–141] , gainezarpen parametroa ξ= 1.0 balioan finkatu dugu. Ondorioz, α berretzailea potentzial harmonikoaren naturaren ( k ) eta guztira erabilitako oinarri funtzioen kopuruaren menpekoa da α=ξ(m+1)2 4x2 0(n,k) . Lan honetan, m=25 jarri dugu. 104 euskarazko laburpena Wigner-en lokalizazioa karakterizatzekotan, sistemaren propietate bi erabili ditugu: gorputz bateko dentsitatea (8.11) eta partikula-hutsune entropia (8.12) . Lehenengoan, φi(r) orbital naturalak dira eta γi koefizienteak korrelazio elektronikoaren garrantziarekin lotzen ditugun okupazio zenbakiak dira [291, 292]. Bigarrenean, aurrekoan bezala, okupazio zenbakiak erabiliz partikula-hutsune entropia agertzen da. Definizioz, adierazpen hori erabiliz lorturiko balioak absolutuki zenbat eta handigoak izan, sistema osoaren karaktere multideterminantala orduan eta handiagoa izango da. ρ(r) = X i γiφ∗ i(r)φi(r)(8.11) S=−X i γilog γi+ (1−γi)log(1−γi)(8.12) Elektroi biz osaturiko sistemak modelizatzeko, CASSCF(2,5) metodoen bidez hainbat k balioentzat spin singlete eta triplete egoeren egitura elektronikoen kalkuluak burutu ditugu. Spin egoera bakoitzarentzat gorputz bateko dentsitateak eskuratu ditugu (2.1 irudia). Ikus dezakegunez, lorturiko dentsitate profila k konfinamendu parametro eta spin egoeraren menpekoa da. Espero bezala, k balio handientzat egoera singleteari dagokion dentsitateak maximo bakarra dauka, ordea, egoera tripleteak (Pauliren esklusio printzipioa dela eta) bi mutur ditu. k balio txikien kasuan, aldiz, spin egoera biek maximo bi dituzte eta antzeko dentsitate profila daukate. Sistema horien izaera multideterminantalaren natura aztertzekotan, hainbat k baliori dagokien egoera tripletearen partikula-hutsune entropiak kalkulatu ditugu. 2.3 irudian ikus dezakegun moduan, k balio handien kasuan partikula-hutsune entropia zerorantza doa. Horrek konfinamendu sendoaren mugan partikula independentearen eredua sistemaren deskribapen ona dela eta lortzen den Fermi-ren gasaren deskribatzeko Slater-en determinante bakar nahikoa dela esan nahi du. Lorturiko kurbaren itxura aurretiaz egindako lanetan lorturiko emaitzekin bateragarria dela [40, 41] adierazi behar dugu. Konfinamendu parametroaren balioa txikitzen joan ahala, karaktere multideterminantalari dagokion entropia handiko egoera lortzen dela ikusi dugu, eta entropiak bere balio maximoa k=5×10−4inguruan izan ondoren, berriz jaisten hasten da. Sistema honi buruzko ezagutza handiagoa izatekotan, oinarri minimodun eredu analitiko bi erabili ditugu. Alde batetik, α berretzailedun eta potentzial harmonikoaren minimoan zentraturiko gausstar bat erabiliz, bi elektroiz okupaturiko orbitala lortzen dugu, horrela egoera singletearen (Fermi-ren likidoa) eredua lortzen dugularik 1Σg=σ↑↓ 0 . Beste aldetik, x0=± 1 /( 4 k)1/3 oreka puntu klasiko bakoitzean α berretzailedun oinarri gausstar bat kokatu eta simetria adaptatuz sistemari dagozkion (2.6) (bikoitia, konbinazio simetrikoa) eta (2.7) (bakoitia, konbinzio antisimetrikoa) orbitalak lortu ditugu. Azken horiek erabiliz, egoera tripletearen (Wigner-en kristala) eredu sinplifikatua lortu dugu 3Πu=σ↑ gσ↑ u . Spin egoera bakoitzarentzari dagokion oinarriei dagozkien gorputz bat eta biko integralak kalkulatuz, egoera bakoitzaren energia espresioak lortu ditugu. Kasu bietan energia k konfinamendu parametroa eta α oinarrien berretzailearen funtzioa dela frogatu ondoren, k bakoitzarentzat energia minimizatu dugu α hura parametro bariazional gisa hartuz. Horrela eginik, egoera bakoitzaren energia optimoaren kurbak lortu ditugu (ikusi 2.4. irudia). Lorturiko emaitzetan oinarrituz, k balio txikientzat, spin egoera bien energiak oso antzekoak izanda, triplearen energia singlearena baino zerbait txikiagoa da. Ostera, 8.3 egindako lanen laburpenak 105 k= 5 × 10 −4 puntuaren inguruan singletea tripletea baino egonkorragoa bilakatzen dela ikusi dugu. Emaitz hauek metodo aurreratuak erabiliz erdietsitakoekin erkatuz, eredu sinple hau lokalizazioa zein k balio inguruan agertuko den aurresateko erabilgarria dela ondorioztatu dugu. Emaitza hauek hartuta, hiru eta lau elektroiz osoturiko sistemtan antzeko joerak agertzen diren ikertu nahi izan dugu. Asmo horrekin, CASSCF( n , 2 n ) (non, n elektroi kopurua den), bakoitzari dagokion gausstarrez eraturiko sarea erbiliz, spin altuko kalkuluak burutu ditugu. Horien irteera balioak erabiliz, hainbat k -ri dagozkien dentsitateak (2.5 irudia) eta partikula-hutsune entropiak (2.6 irudia) kalkulatu ditugu. Horietan ikus dezakegunez, kurba bien itxura antzekoa badute ere, entropia maximoari dagokion k -ren balioa elektroi kopuruaren menpekoa da: zenbat eta elektroi kopurua handiagoa izan, orduan eta txikiagoa izango da k horren balioa. Azterturiko sistema guztiek antzeko joera kualitatiboa dutela nabaritu dugu; ondorioz, Fermi-ren likidoa eta Wigner-en kristalaren arteko trantsizio antzeko prozesu bat S(kmax , n) puntuarekin defini daitekelakoan gaude. Puntu horren posizioa elektroi kopuruaren menpekoa izango da. Batutako emaitzetan oinarrituta zera ondorioztatu dugu: konfinamendua ezartzen duen kanpo potentzial harmonikoaren k parametroaren balio bezain txikientzat sistema eratzen duten elektroiek lokalizatzeko joera dute; horrela, Wigner-en molekula bat eratzen da. Are gehiago, hainbat konfinamendu parametrori dagozkien elektroi-hutsune entropiak kalkulatuz, egoera lokalizatu eta ez-lokalizatuen arteko entropia maximodun trantsizio egoerak aurkitu ditugu. 106 euskarazko laburpena 8.3.2 Sasi bi dimentsio sistemak Dimentsio bateko kasua bi dimentsiotara orokortuz, hainbat elektroiz osoturiko sistemak deskribatzeko, kimika kuantikoan ohikoak diren nukleoetan zentraturiko [142–144] zein espazioko edozein puntu arbitrariotan zentraturiko [138, 145–149] funtzio gausstarrak erabiltzen ditugu. Sistema horiek interes teoriko zein esperimentala dutenez [150–153], egitura elektronikoaren teorian oinarrituko metodoak erabili ahal izateko beharrezkoak diren protokolo konputazioanalak garatzea premiazkoa da. Hala ere, sistema batzuen eskala nanometroaren ordenakoa da, eta horretarako zentro bakar bat aztertzen bada, menpekotasun linealeko arazoak gertatuko dira. Adibide bat ematearren, esperimentalki lorturiko grafenoan oinarrituriko puntu kuantikoek (GQD) eta trantsizio metalen dikalkogenuroetan (TMD) oinarrituriko nanopartikula batzuek 1-7 nm-ko tamaina dute [154–163]. Lan honen bigarren kapituluan, bi dimentsioko potentzial harmoniko simetrikoen bidez konfinaturik dauden hainbat elektroiz osoturiko sistemak deskribatzeko behar diren oinarri gausstar batzuk proposatu ditugu. Gainera, lorturiko oinarri funtzioak potentzial harmonikoaren konfinamendu parametroa k, oinarri funtzio kopuru totala eta gainezarpen parametroarekiko optimioak dira. Horrela, (8.10) ekuazioan agertzen diren s motako orbital gausstarrak sare hexagonal bateko nodoetan kokatu ditugu. Kasuaren arabera, guztira M= 3 g2− 5 g+ 1 oinarri funtzio erabili ditugu, non g≥ 3 zenbaki osoak 3.1 irudian agertzen den patroi hexagonalak dituen zerrenda kopurua adierazten duen. Konfigurazio honetan ondoz ondoko funtzioen arteko gainezarpen integrala S(α , δ) = exp(−αδ2/ 2 ) = exp(−ξ/ 2 ) da, non δ ondoz ondoko funtzioen arteko distantzia den. Definitu dugun ξ=αδ2 gainezarpen parametroaren balioaren arabera, menpekotasun linealak (balore oso txikiak) edo gune ez-jarraituak (balore oso handiak) izan ditzakegu. Aurretiaz buruturiko lanetan oinarrituta [130, 138– 140], lan honetan ξ∈[0.85, 1.15]tarteko balioak aztertu ditugu. Lehen urratsean, potentzial harmoniko isotropiko bidimentsional batean konfinaturik dagoen partikula bateko problemei erreparatu diegu. Horrela, sistema haorei dagokien Hamiltondarra (8.13) ekuazioan agertzen dela eta bere oinarrizko egoeraren energia zehatza E0=k1/2dela atzeman dugu. H(x,y) = 1 2−∂2 x+kx2−∂2 y+ky2=H(x) + H(y)(8.13) Deskribaturiko oinarri funtzioak erabiliz, gorputz bateko Schr¨odinger-en ekuazioa (8.14) adierazpenean agertzen den balore propio orokortuen problema moduan adierazi dugu. Hor, T energia zinetikoa, V energia potentziala, S gainezarpen, E balio propio eta C bektore propio matrizeak dira. Forgatu dugun moduan, energia zinetiko eta potentzialaren matrize horiei dagozkien elementuak gainezarpen matrizearen elementuen menpekoak direnez, guztiak bukle berean definitu ditzakegu. Izatez, erabilitako oinarri funtzioak hiru dimentsionalak direnez, egitan Schr¨odinger-en ekuazioa ebatziz, sasi bi dimentsioko emaitzak erdiesten ditugu. Lorturiko energiak (balore propioak) zuzentzeko, gausstar guztiek berretzaile bera dutenez, zeharkako osagaiaren ekarpenaren ondorioz lortzen dugun gehiegizko energia zinetikoa kedu dugu. Kasu honetan elektroi bakoitzeko α/ 2 energia ekarpena kendu dugu. 8.3 egindako lanen laburpenak 107 (T+V)C=ESC (8.14) Printzipio bariazionalean oinarrituta, (ξ , g , k) balio jakin batzuentzat oinarrizko egoeraren energia ondoz ondoko funztzioen arteko distantziarekiko optimizatuz, hainbat sarreraz osaturiko Ddatu basea lortu dugu. Enpirikoki frogatu dugun legez, era horretan lorturiko Hamiltondarraren balio propioak ez dira bakarrik zehatzak izan, hendekaturiko mailen natura erreproduzitzeko gai ere izan gara. Datu base honetan agertzen diren sarreren eta emaitza analitikoen arteko errore erlatibo maximoa %5ekoa izan da eta hendekaturiko sortentzat kalkulaturiko desbideratze estandar erlatiboa %10−4koa izan da. D datu basea erabilik, osagarriak diren sorta bi lortu ditugu: ikasketa sorta bat L eta aztertze T sorta bat ( D=L∪T , L∩T =∅ ). Horiek erabiliz, bi geruza ezkutuz eta neurona kopuru aldakordunez eraturiko hainbat sare neuronal optimizatu ditugu eta horien funtsa (log k , ξ , g) sarrera moduan emanik log δopt irteera balioa lortzea da. Lorturiko sare optimalak aktibazio funtzio sigmoideak, lehen geruzan sei neurona eta bigarrenean lau ditu. Optimizazioa 10 aldiz gurutzaturiko balioztatzearen bidez egindako errore kuadratikoen minimizazioa burutuz lortu dugu. Emaitza gisa, irteerako balioen estimaturiko errore erlatiboa %2.5ekoa dela ikusi dugu; horren ondorioz, oinarri funtzioen berretzaileen errorea ∆αopt =2ξ δ2 opt ∆log δopt moduan adierazi dugu. Ohartu azken errore hau txikiagoa bihurtzen dela kkonfinamendu parametroaren balioa handitzen den heinean. Lortutako oinarri funtzio optimoek hainbat elektroiz konposaturiko sistemetan agertzen diren korrelazio efektuak deskribatzeko duten gaitasuna frogatzeko asmoz, bi eta hiru elektroiz osoturiko eta spin baxu eta altuko sistemak aztertu ditugu. Horretarako, (8.15) ekuazioan agertzen den eragile Hamiltondarra erabili dugu. H=1 2 n X i=1 (k(x2 i+y2 i)−∇2 i) + n X j>i=1 1 rij (8.15) Efektu horien neurri kuantitatiboa izateko, koordenatu erradialean zehar CASSCF( n ,8) mailan eta UHF mailan kalkulaturiko uhin funtzioen arteko dentsitate diferentzia kalkulatu dugu k∈( 1 × 10 −5 , 1 × 10 −10) balioak erabiliz. Koordenatu erradialaren balio txikientzat uhin korrelatuaren dentsitatea ez korrelatuarena baino txikiagoa dela ikusi dugu. Aldiz, koordenatu erradialaren balio handientzat uhin korrelatuaren dentsitatea ez korrelatuarena baino handiagoa dela ikusi dugu. Efektu hori nabariagoa da k parametroaren balioa handitzen den heinean. Laburbilduz, (20,100) unitate atomikoko tamaina duten sistemak deskribatzeko sare neuronalen bidez lorturiko sare hexagonaletan banaturiko oinarri funtzio gausstarrak lortu ditugu. Etorkizuneko aplikazio gisa, bi dimentsioko grafenoan oinarrituriko puntu kuantikoen tamainako sistemak deskribatzeko erabili ahal izango ditugu. 108 euskarazko laburpena 8.3.3 Hiru dimentsioko Hooke-n atomoa Puntu kuantikoei buruzko ikerketa teorikoetan erabiltzen diren eredu sinple eta egokienetako bat Harmonium edo Hooke-n atomoa deritzona da, non elektroiak potentzial esferiko harmoniko baten bidez konfinatuta dauden [76]. Eredu horiek sistema errealen propietateak simulatzeko doi ditzakegun hainbat parametro dituzte [77, 78]. Esate baterako, gure taldean egindako lan batean kanpo eremu magnetiko bat aplikatuz, bi elektroidun Hooke-n atomo batean gertatzen den singlete-triplete trantsizioa eta horretan korrelazio elektronikoak duen garrantzia aztertu genuen [79]. Hala ere, eredu zehatzak erabiltzeak gure ikerketa bi elektroi sistemetara mugatu zuen. Izan ere, bi elektroiz osoturiko Hooke-n atomoaren kasuan, konfinamendu parametro zehatz batzuentzat baino ( ω2=1 4 , 1 100 . . . ) ez dago soluzio analitikorik[80]. Balio horiek zehatzak badira ere, korrelazio baxu ( ω2→ ∞ ) eta altuko ( ω2→ 0) konfinamenduei buruzko informazioa lortzeko erabilgarriak dira [37, 39, 40, 42–64] [42–44]. Laugarren kapituluan azaldu dugunez, Hooke-n atomoak deskribatzeko bezain egokiak diren oinarriak lortzea premiazkoa da. Sistema Coulomb-arrentzat optmizaturiko aug-cc oinarri nahiko handiak erabiliz elektroi gutxiko sistemak era egokian deskribatzeko gai bagara ere, elektroi kopurua handitzen dugun heinean (hartu 10 elektroidun sistemen adibidea) menpekotasun linealen arazoez gain, ez gara gauza izan oinarrizko egoeraren spin anizkoitztasuna eta, ondorioz, sistemaren betetze egitura era egokian deskribatzeko. ETBS-6S oinarri funtzioak erabiliz (erreferentziazko balioekin alderatuta), behatu dugun errore handiena mHartree ordenekoa izan da. Kapitulu honetan egoera singlete eta tripletean dauden n={ 2, 4, 6, 8, 10 } elektroiz osaturiko ω2= 0.25 konfinamendu parametroa duten Hooke-en atomoak deskribatzeko eta singlete-triplete energia tartea eskuratzeko prozedura bat aurkeztu dugu. Gainera, Yukawa-ren potentzialak erabiliz, elektroi-elektroi elkarrekintzetan pantailatze efektuak duen eragina aztertu dugu, eta, (8.16) ekuazioan agertzen den eragile Hamiltondarra erabiliz, sistema hauen gaineko egitura elektronikoaren kalkuluak burutu ditugu. H=− n X i 1 2∇2 i+ n X i 1 2ω2r2 i+ n X i n X j>i e−λeerij rij (8.16) Energia absolutuentzat balio zehatzak lortzekotan, ω2= 0.25 konfinamendu parametroa erabiliz, hainbat elektroi kopuru desberdin ( n={ 2, 4, 6, 8, 10 } ) dituzten spin egoera singlete eta triplea duten sistementzat optimoak diren oinarri funtzioak lortu ditugu. Horrela, kontraturik gabeko L= 0 eta L= 3 momentu angeluarren arteko eta N geruzadun oinarriak erabili ditugu; horien berretzaileak L eta N zenbakiekiko duten menpekotasuna (8.17) ekuazioan adierazi dugu. ζk LN (ω2) = ω2 2αL,N(ω2)βL,N(ω2)k−1, 1 ≤k≤N(8.17) Azken ekuazio honetan agertzen diren αL,N(ω2) eta βL,N(ω2) parametroak CASSCF( n , 13) eta bi elektroien kasuan FCI mailan lorturiko energiak minimizatuz lortu ditugu; optimizazio prozesu gisa simplex eta Newton-Raphson metodoak erabili ditugu [56, 293, 294]. 8.3 egindako lanen laburpenak 109 Arretaz ikuskatu ondoren, sei elektroi egoera singletean den sistemarentzako lorturiko oinarriek (izenez ETBS-6S) zehaztasunaren eta errendimenduaren arteko oreka onena dutenak direla ondorioztatu dugu. Funtzio horiek trinkotu gabeko 4(SPDF) oinarriak dira eta (8.17) ekuazioan agertzen diren parametro optimoak αL,N= 0.240432 eta βL,N= 1.3021162 izan dira. Horrela, lortzen den berretzaileen segida: 0.2404032, 0.3130329, 0.4076051, 0.5307491 da. Guztira ETBS-6S sorta 80 oinarri funtzioek osatzen dute eta horietariko 68 linealki askeak dira. Bi elektoriz osoturiko sistema Coulomb-iarren emaitzak 4.1 taulan agertzen dira; spin egoera singletearen energia zehatza 2.0 unitate atomikoa (a.u.) dela kontuan izanda, sistema honentzat lorturiko emaitzak gainontzekoenen emaitzen kalitatea kalibratzeko erabili ditugu. Oinarri funtzioen segidan ikusten dugun moduan, FCI mailan: 2.055213 (aug-cc-pVDZ), 2.004107 (aug-cc-pVTZ), 2.000476 (aug-cc-pV5Z) eta 2.000196 (aug-cc-pV6Z) unitate atomiko izan dira. Emaitza horiek literatuan agertzen direnekin bat badatoz ere, singletetriplete energia tartean errorea 0.15 eV-koa da. Ostera, maila beran ETBS-6S oinarriak erabiliz, energia abosuluez gain, tarte horren errorea 0.01 eV-koa izan da; ondorioz, lan honetan garaturiko oinarriak erabiltzea justifikaturik dago. Elektroi kopurua handitu ahala FCI metodoak oso garestiak bilakatzen direnez, CASSCF eta MRMP2 metodoek ETBS-6S oinarriekin lortzen duten zehaztasuna analisatu dugu; 4.2 taulan hainbat espazio aktibo erabiliz lorturiko emaitzak bilduta daude. Ikusten den moduan, era honetan erdietsitako emaitzak FCI metodoaren emaitzen antzekoak dira. Lau elektroiz osoturiko sistementzat, jada, FCI kalkuluak burutzea oso garestia da; bestalde tripletearen oinarrizko egoeran ekarpen handiena duen konfigurazioan 1 p orbitaletan spin bereko bi elektroi daudenez, uhin funtzio multideterminantalak erabiltzea premiazkoa da. 1 p geruzaren hendekapena dela eta, spin egoera bien uhin funtzioak deskribatzeko hainbat konfigurazio biltzen dituzten uhin funtzioak erabili behar ditugu. Gure kasuan, CASSCF(4, m ) eta MRMP2(4, m ) ( m= 4, 5 . . . 13) metodoei esker lorturiko emaitzak 4.3 taulan biltzen dira. CAS(4,13)/aug-cc-pV6Z* eta MRMP2(4,13)/aug-ccpV6Z* metodoak erabiliz, singlete-triplete energia tartea 1.11 eV eta 1.08 eV dira hurrenez hurren; emaitza hauek berez literaturan agertzen direnen antzekoak(1.00 eV)[41] diren arren, CAS(4,13)/ETBS-6S eta MRMPT2(4, 13)/ETBS-6S metodoekin lorturiko energia absolutuen balioen erroreak (erreferentziazko balioekin konparatuz) txikiagoak izanda, singlete-tripletearen emaitza ere hobea dela ikusi dugu. Alegia, singlete-triplete energia artearen balioak 1.06 eta 1.04 eV izan dira. Sei elektroiz osoturiko sistementzat (aurreko kasuaren antzeko eran), tripletearen oinarrizko egoeran ekarpen handiena duen konfigurazioan 1 p bi orbitaletan spin bereko bi elektroi eta beste orbitalean aurkako spina duten bi elektroi daudenez, uhin funtzio multideterminantalak erabiltzea ere beharrezkoa da. Beraz, CASSCF(6, m ) eta MRMP2(6, m ) ( m= 6, 7 . . . 13) metodoei esker eskuraturiko emaitzak 4.4 taulan biltzen dira. Ikusi dogunez, 1 p geruzan beste bi elektroi gehitu arren, singlete-triplete energia tartea aurreko kasuaren antzekoa da; kapitulu honetan optimizaturiko oinarriak erabiliz, 1.03 eV eta 1.00 eV-ko balioak lortzen ditugu (hurrenez hurren) CAS(6,13)/ETBS-6S eta MRMP2(6,13)/ETBS-6S mailetan. Gainera, lortutako emaitzak literaturan agertzen den 0.95 eV-ko emaitzarekin [62] bat datoz. Energia absolutuen zehaztasuna elektroi kopuruarekin txikitzen den arren, estimaturiko singlete-triplete tarteen balioak zuzenak izaten jarraitzen dutela ikusi dugu. Oinarrizko egoera elektronikoaren uhin funtzio monodeterminantala geruza itxikoa bada ere, zortzi elektroidun sistemak ere CASSCF(8, m ) eta MRMP2(8, m ) ( m= 8, 9 . . . 13) 116 euskarazko laburpena natura kobalentea dela ondorioztatu dute [214–216, 218]. 4. kapituluan, lotura kobalentearen deskribatzaileek kanpo potentzial esferiko harmonikoaren indar parametroa ω2 eta nukleoen arteko distantziarekiko R duten menpekotasuna aztertu ditugu. Parametro hauek (8.21) Hamiltondarrean agertzen dira. H=−1 2 4 X i=1∇2 ri−2 2 X J=1 4 X i=1 1 riJ + 4 X j>i=1 1 rij +1 2 4 X i=1 rT iWri+1 R(8.21) Lehen pausuan, 5.8 irudian agertzen diren kobalentzia fase diagramak HF/aug-cc-pVTZ mailan eskuratu ditugu. Etiketa gisa, Laplace-tarrak ∆ρ(rC) eta energia totalaren dentsitateak H(rC) lotura puntu kritikoan duten seinuak erabili ditugu. Era berean, parametro espazioa 500 × 500 sare bat erabiliz diskretizatu dugu [ 0.10, 4.00 ]×[ 1.00, 4.00 ]⊆(ω2 , R) domeinuan eta guztira horren 200 puntu erabiliz (0.1216%) kobelentzia fase diagramak eskuratu ditugu. Maila honetan lortrako ∆ρ(rC) -ren kasuan ikus dezakegunez, balio negatiboak nukleoen arteko distantzia 1.00 eta 1.52 ˚ A tartean dagoenean eta 2.00 kurbatura baino balio handiagoentzat agertzen da. Beste aldetik, H(rC) -ri dagokionez, nukleoen arteko distantzia 1.50 ˚ A baino txikiagoa denean balore negatiboak aurkitu ditugu edozein kurbatura balio erabiliz. Atari balore hori gaindituz, lotura puntu kritikoan ebaluaturiko energia totalaren dentsitatearen seinua aldatu egiten da potentzial kurbatura handitzen dugun heinean; seinu aldaketa hura gertatzeko behar den kurbatura potentzial minimoa nukleoen arteko distantziarekin gorakorra da. Teoria maila honetan, gure erdua erabiliz lorturiko emaitzak hain lan aurrekarietan lorturikoenen antzekoak dira; esate baterako, He2 @ C20H20 sisteman He-He distantzia 1.265 ˚ A izanik ∆ρ(rC) -ren balio positiboa ematen dute [210], gure kasuarekin erkatuz ω2< 2.00. Helio atomoen arteko distantzia 1.60 ˚ A baino txikiagoa den sistementzat ( He2 @ B12N12 eta He2 @ B16N16 )[216] ∆ρ(rC) balio positiboak eta H(rC) balio negatiboak eman dituzte. Intereseko sistema horreri buruz gehiago ikastekotan, 20 × 20 sare uniforme bat eraiki dugu ( 0.00, 1.00 )×( 1.40, 2.6 )⊆(ω2 , R) domeinuan eta puntu bakoitzean CASSCF(4,8)/aug-cc-pVTZ kalkuluak burutu ondoren, ∆ρ(rC) eta H(rC) eskuratu ditugu; horrela, 5.9 irudian agertzen diren sestra kurbak eraiki ditugu. ∆ρ(rC) -ri erreparatuz, domeinu osoan balio positiboak aurkitu ditugu eta bere magnitudea handitu egiten da nukleoen arteko distantzia txikitzen den heinean; era berean kurbatura parametroarekiko ia askea da. Azkenik, domeinu horretan H(rC) -ren balio negatiboak eskuratu ditugu eta euren magnitudea handitu egiten da nukleoen arteko distantzia txikitzen den heinean; azken behaketa hori HF kalkuluetatik eta atomo guztiak erabiliz lorturiko emaitzekin bateragarria dela nabaritu dugu. Laburbilduz, atal honetan ikuspuntu kimikotik interesgarriak diren hainbat aplikazio emanda, ML teknikak fase diagramak eraikitzeko eta fase berriak aurkitzeko tresna erabilgarriak direla ondorioztatu dugu. 8.3 egindako lanen laburpenak 117 8.3.5 Potentzial gausstarrak Konfinamendu potentzial gausstarrek komunitate zientifikoaren arreta erakarri dute. Alde batetik, materia kondentsatuaren teorian matrize erdieroaleetan integraturiko puntu kuantikoen ereduak lortzeko erabiltzen dituzte [233–236]. Berriki bi eta hiru dimentsioko puntu kuantikoei buruzko lanetan hainbat propietateren ereduak lortzeko erabili dituzte, besteak beste: Aharonov-Bohm oszilazioak [237], koherentzia galera efektuak [238], propietate termomagnetikoak [239–242], kanpo eremuen elkarrekintzak [243–248], propietateen menpekotasun topologikoak [249, 250], elkartze kuantikoa [251], eredugintza matematikoa [252– 254], elektroi gutxidun sistemak [255, 256] eta abar. Beste alde batetik, fisika nuklearrean α partikulen arteko elkarrekintzak deskribatzen dituzten Ali-Bodmer potentzialak [257] oraindik ere egitura nuklearreko ereduetan erabiltzen dituzte [258–260]. Azken horietan, potentzial gausstarrak diren moduan erabiltzen diren arren, materia kondentsatuari buruzko lanetan egitura elektronikoaren kalkuluak egiterakoan, potentzial gausstarren ordez horiei hurbiltzen diren potentzial harmonikoak erabiltzen dira. Horiei, Hooke-n atomoan deritze. Horiek kasu batzuetan hurbilketa aproposak izaten badira ere, puntu kuantikoen loturiko egoerak deskribatzerako orduan hainbat mugapen dituzte. Lehenik, potentzial horiek erabiltzeak egitura molekularraren galera dakar; izan ere, hainbat puntutan zentraturiko potentzial harmonien konbinazio lineal ororen emaitza beste zentru bakar bateko potentzial harmonikoa da. Bigarrenez, potenttzial harmonikoei infinitu loturiko egoera eslei diezazkiekegu; hortaz, ionizazioa eta disoziazioa bezalako prozesuak deskribatzea ezinezkoa da. Lan honen 6. kapituluan, hiru dimentsioko zentru bakarreko potentzial gausstarretan konfinaturik dauden n={ 2, 4, 6, 8, 10 } elektroidun sistemak (8.22) ekuazioan agertzen den Hamiltondarra erabiliz aztertu ditugu. Dagozkion bi gorputzetako Yukawa elkarrekintzei dagozkien integralak gure taldean aurretiz egindako lanetatik berreskuratu ditugu [226]. H=−1 2 n X i=1∇2 i− n X i=1 V0e−βr2 i+ n X j>i=1 e−λrij rij (8.22) Potentzial gausstarren bidez konfinaturiko sistemak aztertzeko, beharrezkoak diren N zentru dituen eta (8.23) ekuazioan agertzen den kanpo potentzialari dagozkion matrize elementuak kalkulatu behar ditugu. Gure kasuan, kimika kuantikoan egin ohi denez, matrize elementuak oinarri funtzio gausstarrak erabiliz (8.24) ekuazioan agertzen diren moduan lortu ditugu. Integral hauen kalkulua egiteko, funtzio gausstarren biderkadura teorema (ingelesez Gaussian product theorem) bi aldiz aplikatuz, balio analitiko itxiak eskuratu ditugu. Vext(r) = − N X i=1 V0,ie−βi(r−R0,i)2(8.23) ZG1(α1,RA,l1,m1,n1)G2(α2,RB,l2,m2,n2)e−βi(r1−R0,i)2dr1(8.24) 118 euskarazko laburpena Kalkulaturiko integralen emaitzak egokiak direla eta potentzial hauei buruz gehiago ikastearren, lehendabiziko kalkulu batzuk burutu ditugu. Horien emaitzetan oinarrituta, lorturiko integralen balioa eta horien inplementazio konputazionala zuzenak direla frogatu dugu. Lehenik, {n= 2, 4, 6, 8, 10 } elektroiz osoturiko spin egoera singletea eta tripletean dauden sistemen gaineko CASSCF eta MRMP2 kalkuluak m={ 10, 11, 12, 13 } orbital erabiliz burutu ditugu. Sistema hauek 4. kapituluan agertzen direnekin lotura eratzeko asmoz, V0 parametroaren balio handiak eta zabalera parametro aproposdun potentzial gausstarretan konfinaturiko sistemak aztertu ditugu ω2= 2 βV0= 1 / 4. Hori dela eta, kapitulu hortan aurkezturiko EBTS-6S oinarriak erabiltzea egokia izan da. Ildo berean, V0 parametroaren 30 balio aukeratu ditugu uniformeki banaturiko [ 10.0, 300.0 ] tartean. Hain balio handiak hartuz, funtzio potentzial gausstarraren Taylor-en serie moduan adierazi dugu. Garapen horretan agertzen diren lehen hiru gaiak potentzial harmoniko baliokideei dagozkien elementuak dira; gainera, batezbesteko energia zinetikoak T eta elkarrekintza Coulomb-iarrek Vee potentzial sakonera parametroarekiko menpekotasunik ez dutela hartzen badugu, sistema gausstarraren energia EG eta sistema Hooke-ar baliokidearen energiak EH (6.14) ekuazioaren bidez loturik daude. Adierazpen hori erabiliz eta egitura elektronikoaren kalkuluetatik eskuratutako datuak erabiliz, EH eta lehen zuzenketa anharmonikoa g erregresio linealen bitartez determinatu ditugu (ikusi 6.1 taula). Ikusi dugunez, erregresioen bidez eskuratutako EH balioak 4. kapituluan agertzen diren antzekoak dira; lorturiko errorerik handiena 1 × 10 −5 Hartree-koa izan da. Lehenengo gai anharmonikoari dagokionez, errorerik handiena 4 × 10 −4 unitate atomikoduna izan da. Atzeman dugunez, horren balioa ez da nabari aldatzen espazio aktiboan dauden orbital kopuruarekin, n elektroi kopurua eta spin egoerarekiko, ordea, menpekotasuna nabariagoa da (bi elektroiz koposaturiko sistemen kasua nabarmena da). Korrelazio koefizienteari erreparatuz, kasu guztiak hartuz, bere batezbesteko balioa R2= 0.9991 izan da eta kasurik okerrena CASSCF(8,10)(S)/ETBS-6S kalkuluetatik eratorritakoa R2=0.9623. Azkenik, potentzial gausstarren bidez konfinaturik eta egoera singlean dauden bi elektroiz osoturiko sistemek behintzat loturiko egoera bat dutela kontuan izanik, sistema horiek zein potentzial sakonera baliorentzako disoziatzen diren aztertu dugu ( EG(Vd 0) = 0). Gainera, elektroien arteko elkarrekintzek disoziazio prozesuan duten eragina ikertzeko xedearekin, Yukawa-ren potentzialean agertzen diren λ parametroaren 10 balio hartu ditugu [ 0.10, 1.00 ] tarte uniformean. Era berean, sakonera parametroaren 20 balio hartu ditugu [ 0.50, 1.50 ] uniformeki banaturiko tartean eta egitura elektronikoaren kalkuluak CASSCF(2,13)/ETBS-6S eta MRMP2(2,13)/ETBS-6S mailetan burutu ditugu (ikusi 6.1 irudia). Aurreko kasuan egin dugun antzeko eran, potentzial gausstarra Taylor-en serie moduan garatu dugu, baina kasu honetan bigarren gai anharmonikoa ere hartu dugu; horrela, adierazpen funtzional egokia erabiliz, EH , g1 eta g2 balioak erregresio linealen bidez estimatu ditugu. 6.2 taulan agertzen diren emaitzak kontuan izanik, erregresio tekniken bidez estimaturiko balioek perturbazio metodoek gehitzen duten korrelazio dinamikoarekiko askeak dira azterturiko λ balio guztientzat; horrela, CASSCF eta MRMP2 metodoen bidez lorturiko estimazioak antzekoak direla ohartu gara. Horrela, eskuraturiko sistema Hookear baliokidearen energia EH gero eta txikiagoa da λ handitzen den heinean; hala eta guztiz ere, 4. kapituluan agertzen diren CASSCF/ETBS-6S mailan eskuraturiko emaitzekin alderatuta, era honetan lorturiko energiak batezbeste 0.025 a.u. altuagoak dira. Antzeko neurriko erroreak espero ditugu, izan ere, Hooke-n atomo baliokideen energiak era zehatzean 8.3 egindako lanen laburpenak 119 lortzeko, disoziazio limitean baino, sakonki konfinaturiko sistemak ikertu behar ditugu. Gai ez harmonikoei dagokienez, g1 txikitzen doa λ handitzen den heinean. Hipotesi nagusia: elektroi-elektroi elkarrekinztak ahultzen diren heinean, korrelazio efektuak ere itzaltzen joango dira eta elektroiak potentzial putzuaren minimotik hurbil aurkitzeko aukera handitu egiten da; hortaz sistema osoa ”harmonikoagoa” da eta g1 -en magnitudea txikitu egiten da. Bestalde, g2 bigarren gai anharmonikoa hazi egiten da λ= 0.6 punturaino eta gero txikitzen hasten da. Disoziazio atarien balioa ahalik eta zehatzenak eskuratzekotan, Taylor-en garapenaren gaien seinuak alternatuak direla frogatu ondoren, funtzioa Stiljies motakoa dela ohartu gara. Hori horrela, Pad´e-ren sekuentzia nagusiak hartuta ( P1 1(V−1 0) eta P1 2(V−1 0) ) disoziazio sakonera limitearen goi eta behe mugak determinatu ditugu. 6.2 irudian ikus dezakegunez, pantailatze parametroa zenbat eta handiagoa izan, elektroi-elektroi elkarrekintzak orduan eta ahulagoak izango dira eta potentzialak ez du zertan hain sakona izan behar partikulak konfinaturik (energia negatibodun egoera) mantentzeko. 120 euskarazko laburpena 8.4 EMAITZA NAGUSIAK 8.4.1 Konfinamendu harmonikoa ikertzeko protokolo konputazionalen garapena Tesi honetako 3., 4. eta 5. kapituluetan, sasi bat, sasi bi eta hiru dimentsiodun potentzial Harmonikoetan konfinaturiko elektroi gutxidun sistemak aztertzeko konputazio protokoloen garapena aurkeztu dugu. Alde batetik, sasi-bat eta sasi-bi dimentsiotako sistemak aztertzeko, hiru-dimentsioko s motako oinarri funtzio gausstar banatuak erabili ditugu. Lehenengoentzat behar adina funtzio banatu ziren uniformeki banatutako sare batean zehar. Funtzio horiei dagozkien berretzaileak α eta zentruak {ri} , k konfinamendu parametroaren menpekoak izanik, potentzialaren informazioa integraturik dute; bestalde, ondoz-ondoko funtzioen arteko gainezarpen integralaren balioa S(α , δ) = exp(−αδ2/ 2 ) = exp(− 1 / 2 ) ezarri dugu. Bigarrenarentzat, funtzio gausstarrak sare hexagonal batean banaturik daudela, ondoz ondoko oinarri funtzioen arteko distantzia δ , gainezarpen parametroa ξ , konfinamendu parametroa k eta oinarri-funtzio konpuru totala emanik dagozkien α berretzaileak bariazionalki optimizatu ditugu. Zenbait optimizazio egin ditugu konfinamendu parametroen hedadura handia azaltzeko. Partikula bakarraren eredutik lorturiko emaitza analitikoak erabiliz, sare neuronal ereduak optimizatu eta trebatu genituen emaitza horiek edozein baliotara hedatzeko helburuarekin. Konfigurazio horiek erabiliz eredu horientzat errore txikiak lortu ez ezik, metodo multi-erreferentzialei esker konfinamendu txikiko sistemetan agertzen den Wigner-en lokalizazioa modu egokian errepresentatu ere egin dugu. Beste alde batetik, lau geruzaz osoturiko zentru bakarreko oinarri funtzioak (izenez 4SPDF ) k=ω2= 1 / 4 konfinamendu parametrodun Hooke-n hiru-dimentsioko atomoentzat optimizatu ditugu. Hainbat elektroi kopuru ( n={ 2, 4, 6, 8, 10 } ), spin egoera (singlete eta triplete) eta pantailatze parametroren (λ={ 0.0, 0.2, 0.4, 0.6, 0.8, 1.0 }) optimizazioak burutu ondoren, oinarri unibertsal optimoa sei elektroiz eta egoera singleteko sistemarako lortutakoa zela ondorioztatu dugu (izenez ETBS-6S). Oinarri horiek baliatuta, sistema guztien singlete eta triplete spin egoeren energia absolutuak lortuz, aurretiz argitaraturiko emaitzekin bat datozela ikusi dugu. Horrez gain, lortutako singelete-triplete diferentzia balioak zehatzak eta sistemaren Aufbau egitura zuzena ere badela egiaztatu dugu. 8.4.2 Ziurgabetasun laginketen metodoen inplementazioa fase diagramak lortzeko Bosgarren kapituluan, ETBS-6S oinarriaren optimizazio-prozesua eta zehaztasuna deskribatu ditugu. Aipatu bezala, horiek medio direla, Hooke-n atomo esferikoen egitura elektronikoko kalkuluen bidez lortzen den Aufbau egitua 1 s < 1 p < 1 d dela ikusi dugu. Betetze-egitura honetan oinarrituz, lau eta sei elektroiz osaturiko Hooke-n atomo esferikoen oinarrizko egoeraren spin multiplizitatea tripletea dela berehala ikusi dugu. Hala eta guztiz ere, simetria esferikoa apurtuz gero, p orbitalak ez dira gehiago hendekaturik egongo, eta talde bitan bereiziko dira: {px , py} eta {pz} ; horrela, oinarrizko egoeraren spin multipliz- 8.4 emaitza nagusiak 121 itatea simetria esferikoa nola hausten den eta sistema konposatzen duen elektroi kopuruaren menpekoa izango da. Hooke-n atomo ez-esferikoen oinarrizko egoeraren spin aniztasuna (singletea edo tripletea) zein baldintzatan ezartzen den ulertzeko, 6. kapituluan erdi-gainbegiraturiko ikaskuntzan eta ziurgabetasun laginketa metodoetan oinarritutako makina-ikaskuntza prozedura bat proposatu dugu. Lehen pauso gisa, proposaturiko prozesua ilustratzeko, puntu eutektiko bakar bat duten ereduzko solido-likido sistema batzuen fase diagramak lortu ditugu. Ikus dezakegunez, ziurgabetasun laginketa teknikei esker, algoritmoa iterfase guztiak lagintzeko gai da, bereziki, puntu eutektikotik hurbil dauden puntuak. Hooke-n atomo anisotropikoei dagokienez, x eta y norabideetan ω2 x,y konfinamendu parametroa eta z norabidean ω2 z konfinamendu parametroa ezarri ditugu. Gainera, elektroielektroi elkarrekintza modulatzen duen λ Yukawa parametroarekin ere hainbat froga egin ditugu. Adierazitako laginketa prozesua gauzatu ondoren, kasu esferiko guztientzat (ω2 x,y=ω2 z , ∀λ) oinarrizko egoeraren spin multiplizitatea tripletea izan dela atzeman dugu. Bestalde, ω2 z> ω2 x,y den kasuetan, lau elektroidun sistementzat oinarrizko egoeraren spin multiplizitatea singletea den bitartean, sei elektroidun sistementzat tripletea izaten jarraitzen du. Kontrara, ω2 z< ω2 x,y den kasuetan, sei elektroidun sistementzat oinarrizko egoeraren spin multiplizitatea singletea den bitartean, lau elektroidun sistementzat tripletea izaten jarraitzen du. Sistema eredugarri horiek etorkizunean erabili ahal izango dira elektroi kopuru handiagoek osatutako sistemak aztertzeko. Azkenik, metodoaren erabileraren garrantzia ilustratzen duen beste adibide bat emateagatik, potentzial esferiko harmonikoetan konfinaturiko helio-dimeroen izaera kobalentea Bader-en teorian agertu ohi diren lotura deskribatzaileak erabiliz aztertu dugu. Kasu honetan, helio atomoen arteko lotura kimikoaren izaera kobalentea neurtu dugu, lotura puntu kritikoan kalkulatutako dentsitatearen Laplacetarraren eta guztizko energia-dentsitatearen zeinuak erabiliz. Gainera, potentzialaren konfinamendu parametroa eta distantzia internuklearra, fase-diagrama eraikitzeko kanpoko parametro gisa erabili ditugu. Egoera kobalenteen eskualde ezberdinak iragarri arren, kualitatiboki deskribatzaile biek nukleoen arteko distantzia txiki eta konfinamendu balio handietan egoera kobalenteen existentzia aurresaten dute. 8.4.3 Konfinamendu gausstarrei dagozkien gorputz bateko integralak inplementatu eta aurretiaz lorturiko emaitzak hedatu Orain arte ikertutako sistemetan, elektroiak potentzial harmonikoen bidez konfinaturik egon dira. Eredu horiek puntu kuantikoen eta atomo artifizialen loturiko egoerak deskribatzeko lehen hurbilketa gisa erabil ditzakegun arren, ez dira kapaz sistema errealetan garrantzitsuak diren hainbat ˜nabardura deskribatzeko. Lehenik, hainbat puntutan zentraturiko potentzial harmonikoen konbinazio lineal bat oraindik potentzial harmoniko bat denez, zentru anitzetatik eratorritako egitura molekularra behatzea ezinezkoa da. Bigarrenez, potentzial harmonikoek infinitu egoera lotu dituzte; hori dela eta, ionizazioa eta disoziazioa bezalako prozesuak deskribatzeko erabilezinak dira. 122 euskarazko laburpena Lan honen zazpigarren kapituluan, konfinamendu gausstarren garrantzia eta propietate orokorrak aipatzen ditugu; besteak beste (potentzial harmonikoak ez bezala) egitura molekularra eta loturiko egoera kopuru finituak dituztela. Horretaz gain, beharrezkoak diren gorputz bateko integralen kalkuluaren formulazioa adierazi dugu. Potentzial horiek erabiliz, lehendabiziko emaitza batzuk emateko asmoarekin, kasu ilustratzaile bi aurkeztu ditugu. Alde batetik, sakonki konfinaturiko ( V0 sakonera parametroaren balio handiak) elektroi kopuru bikoitiz ( n={ 2, 4, 6, 8, 10 } ) eraturiko sistemen singletetriplete energia tartea CASSCF( n , m )/ETBS-6S eta MRPT2( n , m )/ETBS-6S mailetan m={ 10, 11, 12, 13 } orbital aktibo erabiliz kalkulatu eta aurretiaz Hooke-n atomoentzat lorturiko emaitzekin alderatu ditugu. Taylor-en serieak eta erregresio linealak tarteko, Hooke-n atomo baliokidearen energia eta lehen gai anharmonikoa erregresiotik lorturiko parametroak erabiliz lortu ditugu. Era horretan lorturiko Hooke-n atomo baliokidearen energiak eta laugarren kapituluan agertzen direnak oso antzekoak direla ikusi dugu. Beste alde batetik, Yukawa potentzialen bidez elkarrekitzen eta spin egoera singletea duten elektroi biz osoturiko sistemaren disoziazio mugak ikertu ditugu. Kasu honetan, Taylor-en serieak erabili beharrean, konbergentzia azkarragoa duten Pad´e-ren sekuentzia nagusietara jo dugu; horrela, objektu horien propietateetan oinarrituta, hainbat pantailatze parametrori dagozkien goi eta behe disoziazio mugak lortu ditugu. 8.5 ondorio nagusiak 123 8.5 ONDORIO NAGUSIAK Lan honetan potentzial estatikoen bidez konfinaturiko elektroi sistemei buruzko aurrerapen batzuk jaso dira. Lehenik, bat, bi eta hiru dimentsioko potentzial harmonikoei esker, hainbat spin zenbaki dituzten elektroi gutxiz konposaturiko sistemak aztertzeko konputazioprotokoloak garatu ditugu. Zehazki, batetik, optimizazio teknika klasikoen bidez, zentru bakarreko eta dimentsio baten eta bitan banaturiko oinarri funtzio gausstar egoki batzuk lortu ditugu. Beste alde batetik, oinarri funtzio horiek CASSCF eta MRPT2 metodo multi erreferentzialak erabiliz: konfinamendu ahulean gauzatzen den Wigner-en lokalizazioa eta singlete-triplete energia tarteak modu egokian deskribatzeko gai izan gara. Hooke-n atomo esferikoen kasuan, sistemaren Aufbau egitura hartuta, oinarrizko egoeraren spin anizkoitztasuna elektroi kopuruaren menpekoa da. Halere, simetria esferikoa apurtu egiten da ( ω2 z=ω2 x,y ) oinarrizko egoeraren spin multiplizitatea elektroi kopuruaren menpe egoteaz gain, simetria hori hautsi izan dugun eraren menpekoa ere bada. Elektroi kopuru jakin bat hartuta, erdi gainbegiraturiko eta ziurgabetasun lagineketa teknikak erabiltzen dituen garaturiko makina ikasketa metodo baten bidez, konfinamendu eta pantailatze parametroen menpekoak diren oinarrizko egoeraren spin multiplizitatearen fase diagramak eraikitzea lortu dugu. Gainera, metodo hori kimikoki esanguratsuak diren beste sistema gehiago ikertzeko (solido-likido nahaste binarioak eta lotura kimikoaren naturaren analisia) erabilgarria dela frogatu dugu. Potentzial harmonikoen luzapen gisa, zentru anitzetatik eratorritako propietate molekularrak, disoziazio prozesuak eta ekarpen anharmonikoak biltzen dituzten konfinamendu potentzial gausstarrei sarrera eman diegu. Beharrezkoak diren gorputz bateko integralak GAMESS-US softwear-ean inplementatzeaz gain, baliokideak diren sistema Hooke-ondarrekin alderatuz lorturiko emaitzei erreparatuz potentzial bi horien arteko lotura ere frogatu dugu. Etorkizun hurbilean, lan honetan lortutako emaitzak eta garaturiko metodoak konfinamendu erregimen gehiagotara eta sistema errealistagoetara hedatuko ditugu. 124 euskarazko laburpena 8.6 ETORKIZUNERAKO LANAK Laburbilduz, lan honetan sasi-bat, sasi-bi eta hiru dimentsiotan potentzial harmonikoen bidez konfinaturiko elektroi sistemak deskribatzeko beharrezkoak diren protokolo konputazioanalak garatu ditugu. Horretarako, makina ikaskuntzan oinarrituriko hainbat teknika erabili ditugu beharrezkoak diren oinarri funtzioak optimizatzeko eta egitura elektronikotik eratorritako propietateei dagozkien fase diagramak irudikatzeko. Azkenik, konfinamendu potentzial gausstarrei dagozkien gorputz bakarreko integralak lortu eta inplementatu ondoren, potentzial harmonikoarekin lorturiko emaitzekin daukaten lotura ere ezarri dugu. Lortutako ezagutza beste sistema batzuetara hedatzeko helburuz, eta gure taldearen esperientzian oinarrituta, lan honetan garaturiko ereduak dopaturiko kluster endoedrikoetan aplikatzea kontsideratu dugu. Orain, proiektu honi buruzko bi gai nagusiak aurkeztuko ditugu: atomo artifizial eta superatomoekin loturik dauden dopaturiko kluster endoedrikoak eta potentzial estatikoei esker konfinaturiko sistema kuantikoak. Bi alorretan eskuratutako lan eta emaitza batzuk eztabaidatu ondoren, bi diziplinen arteko zubi bat ezarriko dugu eta gure ikuspegia aurkeztuko dugu. Alde batetik, klusterrak materia kondentsatuaren eta molekulen artean daude, eta horrek propietate elektroniko interesgarriak izatea eragiten du. Sistema horien artean, hutsik dauden nanokluster esferikoak aurki daitezke, esate baterako aintzindaria izan zen fulereno 60 lantanorekin dopaturiko klusterra La@C 60 [270], antzekoak diren beste hainbat [271, 272] eta potentzialki erloju atomiko gisa erabil daitezkeen fulereno nitrogeno klusterrak [273]. Fulerenoan oinarritutako espeziez aparte, ”superatomo” izendun hainbat sistema ere badaude[274–277], besteak beste, Al 12 kutxen kasuan, dopatzailearen naturaren arabera, sistema osoa superhalogeno (boroarekin dopatuz), superalkalino (fosforoarekn dopatuz), superkalkogeno (kaltzioarekin dopatuz) edo 40 elektroidun sistema egonkor gisa (siliziorekin dopatuz) har dezakegu [278]. Beste sistema interesgarri batzuk kluster erdieroaleetan oinarritutakoak dira, zeinetarako hainbat propietate doikor (xurgapena, emisioa eta fotolumineszentzia) hauteman diren [279–285]. Sistema superatomikoetan, propietate ”atomikoak” egitura elektroniko osoaren portaera kolektibotik sortzen dira; haatik, kluster erdieroaleetan konfinaturiko atomoen kasuan, sistema osoaren propietateak elementu dopante isolatuetan behatutakoen oso antzekoak dira [286–290]. Beste alde batetik, konfinamendu potentzial gausstarrak egitura elektronikoari buruzko bai lan teorikoetan [252–256], bai materia kondentsatuari buruzko lan aplikatuagoetan erabili izan dituzte; hala nola: puntu kuantikoak [233–237], propietate termomagnetikoen ikerketa [239–242] eta LASER eta eremu elektrikoak eta materiaren arteko elkarrekintzak aztertzeko [243–248]. Potentzial horiek egitura elektronikoko metodoetan lorturiko emaitzak kalibratzeko eta horien kalitatea neurtzeko erabiltzen diren potentzial harmonikoen antzekoak diren arren [36, 37, 41, 80, 261–266], haien arteko desberdintasun nagusia lehenengoa egitura molekularrak eta loturiko egoera finituak deskribatzeko gai dela da (6. kapitulua). Bi kontzeptu horiek (dopaturiko kluster erdieroaleak eta potentzial gausstarrak) kontuan inzanda, lehen urrats bezala, potentzial gausstarretan konfinatuta dauden metal alkalinoen eta halogenoen ionizazio energiak eta afinitate elektronikoak ikertzea gustatuko litzaiguke. Atomo askeen ionizazio energia, okupaturiko energia garaienaren orbitala eta lehen ordeneko perturbazio metodoa (eta Helmann-Feyman teorema aplikatuz) erabiliz, 8.6 etorkizunerako lanak 125 energia horien hurbilezko balioak erdietsi ditugu eta euren balioak kalkulu esplizituetatik lorturiko emaitzekin erkatu ditugu. Sistema endohedrikoentzako hasierako emaitza batzuk V0 sakonera eta β zabalera parametroak dituen potentzial gausstar baten bidez Z zenbaki atomikodun eta n elektroidun atomo baten Hamiltondarra (8.25) ekuazioaren bidez adieraziko dugu. H=−1 2 n X i=1∇2 i−Z n X i=1 1 ri + n X j>i 1 rij −V0 n X i=1 e−βr2 i(8.25) Uhin funtzio monodeterminantalak hartuta, hurbilketa adiabatikoa (Koopmans-en teorema) eta Helmann-Feyman teorema aplikatuz, ionizazio potentzialak sakonera parametroarekiko duen menpekotasunari dagokion eta n -garren orbital atomikoaren menpekoa den espresio hurbildu bat lortu dugu: ∂I ∂V0 =−*Ψion  n−1 X i=1 e−βr2 iΨion++*Ψatom  n X i=1 e−βr2 iΨatom+ ≈Dϕ(rn)e−βr2 nϕ(rn)E Horrela, V0 parametroaren balioa txikia dela hartzen badugu, Helmann-Feynman teorema erabiliz eskuraturiko espresioa V′ 0∈[ 0, V0] tarte txikian integratuz gero, ionizazio energiaren espresio hurbildua lortu dugu (ikusi (8.26) ). Formulazio alternatiboa: V0 parametroaren balio txikiak hartuz, lehen ordeneko perturbazio zuzenketa erabiliz emaitza bera eskuratu dugu. Jakina, hurbilketa horretan dagokion orbital atomikoaren (eta era orokorrean uhin funtzio osoaren) forma potentzial paratroekiko askea dela aintzat hartu dugu; kasu orokorrean hori ez da zuzena Hooke-n atomoen kasuan ikusi dugun moduan (4. eta 5. kapituluak). Ondorioz, ω2= 2 βV0 moduan definituriko kurbatura parametroa etorkizuneko ikerketetan kontuan izateko ezaugarri adierazgarria izango dela uste dugu. I≈I0+V0Dϕ(rn)e−βr2 nϕ(rn)E(8.26) Antzeko estrategia jarraituz, n elektroidun atomo neutro bat eta honi dagokion n+ 1 elektroidun anioia hartuz, A afinitate elektronikoa anioiaren ionizazio energia baina seinu negatiboa duen kantitate bezala definituz, (8.27) ekuazioan agertzen den hurbilketa lortu dugu. A≈A0−V0Dϕ(rn+1)e−βr2 n+1ϕ(rn+1)E(8.27) Adierazpen bietan ( (8.26) eta (8.27) ) ikus dezakegun moduan, V0 parametroaren balio adin txikiak hartuta, β parametroaren balioa ere txikia bada, potentzial gausstarrak sistema 132 bibliography [73] L. Vandersypen and M. 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