Study of the electronic and rovibronic structure of the X 2Σ+, A 2Π, and B 2Σ+ states of AlO
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Study of the electronic and rovibronic structure of the X 2Σ+, A 2Π, and B 2Σ+ states of AlO Andrei T. Patrascu, Christian Hill, Jonathan Tennyson, and Sergei N. Yurchenko Citation: J. Chem. Phys. 141, 144312 (2014); doi: 10.1063/1.4897484 View online: https://doi.org/10.1063/1.4897484 View Table of Contents: http://aip.scitation.org/toc/jcp/141/14 Published by the American Institute of Physics Articles you may be interested in The microwave spectrum of the AlO radical The Journal of Chemical Physics 92, 2146 (1990); 10.1063/1.458005 Laser fluorescence study of AlO formed in the reaction Al + O2: Product state distribution, dissociation energy, and radiative lifetime The Journal of Chemical Physics 62, 1824 (1975); 10.1063/1.430710 Ground and excited states of the diatoms CN and AlO The Journal of Chemical Physics 61, 1274 (1974); 10.1063/1.1682049 Band strengths for electric dipole transitions from ab initio computation: Lio AlO The Journal of Chemical Physics 58, 4412 (1973); 10.1063/1.1679002 Chemiluminescence of AlO The Journal of Chemical Physics 63, 1963 (1975); 10.1063/1.431530 Ab initio dipole moment functions for the X 2Σ+ and B 2Σ+ states of AlO The Journal of Chemical Physics 77, 6083 (1982); 10.1063/1.443852
THE JOURNAL OF CHEMICAL PHYSICS 141, 144312 (2014) Study of the electronic and rovibronic structure of the X 2+, A 2, and B 2+states of AlO Andrei T. Patrascu, Christian Hill, Jonathan Tennyson, and Sergei N. Yurchenko Department of Physics and Astronomy, University College London, Gower Street, WC1E 6BT London, United Kingdom (Received 1 June 2014; accepted 29 September 2014; published online 14 October 2014) The electronic structure of the X2+,A2, and B2+states of aluminum monoxide (AlO) are studied via ab initio multi-reference configuration interaction calculations. Core correlation corrections, several basis sets, and active space choices are considered. Angular momentum and spin-orbit coupling terms are obtained at different levels of theory. The resulting ab initio curves are used to solve the associated rovibronic problem for the total angular momentum Jup to 112.5 and then also refined by fitting to the experimental wavenumbers available in the literature, reproducing them with the root-mean-square error of 0.07 cm−1. Theoretical rovibronic energy levels of AlO in its X2+, A2, and B2+electronic states are presented including those from the X−Bblue-green system. © 2014 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4897484] I. INTRODUCTION An accurate spectral description of aluminum monoxide (AlO) has become particularly important after the recent discovery of AlO spectral lines in the atmosphere of a new class of the nova-like stars,1of which the most prominent examples are probably V838 Mon and V4332 Sgr.2–4These two objects defined a new type of eruptive variables called intermediateluminosity red transients and the observational data showed the intense presence of the A−Xsystem of the AlO radical. Indeed, this A−Xband is also found to be fairly prominent in a variety of cool, oxygen rich stars4while B−Xlines have been observed in sunspots.5 AlO emissions are also observed in the Earth’s atmosphere, both naturally6and as a result of experiments.7It is also a major constituent of rocket exhausts; its spectrum is important for monitoring interaction of these gases with the atmosphere.8,9It is also used for monitoring reactions involving aluminum and oxygen in the laboratory.10–13 In 1927, Pomeroy14 reported for the first time the resolved rotational structure of the 1 −0, 0 −0, and 0 −1 bands of the blue-green band system of AlO which involved transitions between the B2+and X2+states. In 1937, Sen15 used higher resolution to reanalyze both the rotational structure of these and a few more bands. He showed that spindoubling exists in both the upper B2+state and the ground X2+state, and not just in the X2+state as was assumed previously.14 In 1957, Lagerqvist et al.16 analysed many more bands in the B−Xsystem of AlO and gave precise values for the rotational constants Band Dfor levels up to v=3 in the Bstate and up to v=5intheXstate; they also obtained the spin-doubling constants γfor these levels. Mahieu et al.17 excited the blue-green system of AlO in a hollowcathode discharge cooled by liquid nitrogen. From the difference of the intensity ratios in the P2(N), P1(N) and R2(N), R1(N) rotational lines with low N, and also from theoretical considerations, they showed that the spin-doubling constant γin the Bstate is negative, contrary to the assumptions of the earlier studies.14,16,17 However, they confirmed that the value of γin the Xstate is positive. Later, the rotational structure of ten more bands of the blue-green system, namely, 2 −0, 3−1, 4 −2, 5 −3, 6 −4, 7 −5, 4 −1, 5 −2, 6 −3, and 7−4 were analysed.18,19 Singh and Saksena20 measured line frequencies of unblended intense rotationally resolved lines with an accuracy of 0.05 cm−1. Since that time a series of studies have extended both the range and accuracy of the measured vibronic spectrum.21–23 Herzberg and Huber24 listed seven electronic states of AlO, X2+,A2,B2+,C2,D2+,E2, and F2+, as having been identified experimentally. A combined analysis of the A2→X2+and B2+→X2+band systems of AlO based on an experimental emission spectrum is given by Launila and Berg25 who provide an extensive list of measured transition frequencies. Several theoretical studies of the electronic states of aluminium monoxide have been performed. Ab initio and density functional methods were employed by Orlova and Goddard26 to describe the thermochemistry of AlO. The X2+,B2+, D2+, and F2+states of AlO were studied via ab initio configuration interaction (CI) calculations and associated molecular vibration calculations.27–29 Very recently, Liu and Shi30 used multi-reference (MR) CI to obtain spectroscopic constants for several low-lying doublet and quartet states of AlO. Ab initio state-averaged complete-active-space self-consistent-field (SA-CASSCF)/Multi-Reference Configuration Interaction (MRCI) calculations were performed by Lengsfield, Liu, and co-workers27,31 for the blue-green system B2+−X2+. The most recent ab initio study of the Xand Bstates, which characterized the potential energy curves (PECs), vibrational energies, spectroscopic constants, dipole moments, using MRCI wavefunctions, was reported by Honjou.29 We compare our results with the theoretical work of Honjou29 and that of Zenouda et al.32 in which extended CASSCF (averaging 12 electronic states) and MRCI approaches were used. 0021-9606/2014/141(14)/144312/10/$30.00 © 2014 AIP Publishing LLC141, 144312-1
144312-2 Patrascu et al. J. Chem. Phys. 141, 144312 (2014) Our goal is to generate all components of the Schrödinger equation plus diagonal and off-diagonal effects due to relativity (see below) required for an accurate description of the electronic systems involving the three lowest electronic states X2+,A2, and B2+of AlO, including the potential energy curves, spin-orbit curves (SOCs), and electronic angular momentum curves (EAMCs), EAMCs are obtained as matrix elements of the electronic angular momentum operator, ˆ L, which together with SOCs participate in the couplings between the electronic states. The ab initio PECs, SOCs, and EAMCs are then refined by fitting to the experimental data for 27Al16O due to Saksena et al.,23 and Launila and Berg25 with the total angular momentum quantum number Jranging up to 112.5. This is the first step towards providing a comprehensive line list for hot AlO as part of the ExoMol project.33 This project has already provided line lists for BeH, CaH, MgH,34 SiO,35 and NaCl and KCl,36 but AlO is the first system for which it is necessary to consider strongly coupled PECs. The paper is structured in the following way. Section II presents the ab initio methods, where special attention is given to their effectiveness in the calculation of potential energy curves, the selection of the basis sets and of the active space. The methods used to solve the rovibronic problem are described in Sec. III. The results of the refinement of the ab initio potential energy and coupling curves by fitting to the experimental data from the literature are given in Sec. IV. II. CONFIGURATION INTERACTION CALCULATIONS Increase in computational power has allowed many improvements of the theoretical results obtained through ab initio methods. This is in part also due to the fact that new methods have been implemented and became standard tools in quantum chemistry. In this paper, we use the multi-configuration self-consistent field (MCSCF) method in conjunction with the subsequent MRCI method and the following basis sets: aug-cc-pV(X)Z (X=Q,5,6) for the main calculations and aug-cc-pCVQZ for the core-correlation adapted calculations. These calculations were performed using the MOLPRO electronic structure package37,38 and the Gaussian basis sets defined therein. In this paper, we shall follow the MOLPRO notation in order to classify different electronic configurations. In this notation, the total number of closed shell orbitals is separated into orbitals belonging to different representations of the C2v point symmetry group. The six closed shell orbitals considered here are grouped in the symmetry representations of C2vas follows: four a1 orbtials, one b1orbital, one b2orbital, and no a2orbitals. The five further orbitals that constituted the active space had symmetries: three a1orbitals, one each of b1,b2and no a2orbitals. This orbital set is called (7,2,2,0) or (7,2,2). Molecular properties determined here are presented in Tables Iand II where they are compared to those from the theoretical studies of Honjou,28,29 Zenouda et al.,32 and the experimental work of Launila and Berg.25 Although the accuracy of the predictions obtained is comparable to that of TABLE I. Bond lengths, Re(Å), for the first three electronic states of AlO computed. State Ab in.aAb in.bRefinedaExpt.c X2+1.6341 1.623 1.6189 1.6179 A21.7831 1.777 1.7691 1.7678 B2+1.6838 1.677 1.6681 1.6670 aThis work. bReference 32. cReferences 24 and 39. previous theoretical work, such as that of Honjou,28,29 it is still far from the experimental accuracy. We have also tested the configurations in the form (8,3,3) and (9,4,4) and the augmented correlation consistent polarized valence-only basis sets up to 6Z. Inclusion of relativistic and core corrections leads to an overall and significant decrease in the energy of the system, but this mostly amounts to a global shift of the potential energy curves and gives corrections that are small with respect to the experimental error. In particular, Table II gives results obtained using a variety of different high-level ab initio models. None of the results even begin to approach spectroscopic accuracy. There is an additional problem in that the more demanding models gave off-diagonal terms, relativistic couplings, or transitions dipole moments, which were far from smooth. It is not really possible to work with such curves. For this reason, the results of Table II were all obtained using our final, refined couplings given below. We found that the improvements on the potential energy curve with 5Z and 6Z basis sets do not justify the increased computing time. The same is true for the increased active spaces we tested. The final choice of the active space was (7,2,2) and an aug-cc-pV4Z which was optimal in terms of performance and smoothness. The computed PECs, SOCs, and EAMCs are presented in Figs. 1and 3. As already mentioned, smoothness of the curves is far from guaranteed in these calculations.41 Here, they are the result of a careful choice of the starting point. We started generating the potential energy functions and implicitly all other geometry dependent properties at the shortest internuclear distance. We continued by successively increasing the bond length Rwhile keeping as an initial configuration the one generated at the previous position. We varied the bond length in the range R=1.2–2.74 Å. The off-diagonal ab initio couplings and transition dipoles as obtained by MOLPRO are often affected by the ambiguity of phases associated with electronic wavefunctions, which have an arbitrary overall sign for any given geometry. This problem is illustrated in Fig. 2, where raw MOLPRO data for different couplings are shown; a transition dipole moment is also shown for comparison. This ambiguity must be eliminated by a consistent choice of the phases for all the matrix elements. To this end, we chose the phase at the reference geometry of R=1.84 Å and corrected all MOLPRO matrix elements to be smooth for all geometries, see Fig. 2. The corrected ab initio SOCs and EAMCs are also shown in Fig. 3.
144312-3 Patrascu et al. J. Chem. Phys. 141, 144312 (2014) TABLE II. J=0 energy term values and electronic excitation energies obtained using different high-level ab initio treatments compared to the refined values (this work) observed values.39,40 All data in cm−1. All calculations at MRCI level.a State Vib pVQZ (8,3,3)bpVQZ(8,3,3)MVDcpV5Z(8,3,3)DKdpVQZ(9,4,4)eObs. Refined X21 949.98 948.70 957.04 944.56 965.45 965.44 X22 1887.62 1884.98 1899.06 1874.18 1916.86 1916.85 X2b3 2808.61 2804.54 2825.96 2788.94 2854.24 2854.21 X24 3711.43 3706.17 3737.70 3689.03 3777.50 3777.50 X25 4599.10 4593.61 4634.29 4574.71 4686.71 4686.66 X26 5464.35 5469.44 5515.82 4684.94 5581.93 5581.91 X27 6346.30 6332.07 6382.48 5396.53 6463.13 6463.04 A21 724.42 711.58 ... ... 720.94 720.82 A22 1410.20 1414.35 ... ... 1432.88 A23 2106.85 2108.41 ... ... 2137.03 A24 2793.83 2793.84 ... ... 2813.05 A25 3472.25 3470.75 ... ... 3512.73 3511.15 A26 4143.50 4139.24 ... ... 4189.08 A27 4799.53 4799.44 ... ... 4857.90 B21 847.02 847.03 ... ... 863.11 862.99 B22 1688.34 1688.35 ... ... 1718.85 1718.87 B23 2525.03 2525.04 ... ... 2567.28 2567.36 B24 3355.99 3356.01 ... ... 3408.41 3408.44 B25 4178.57 4178.59 ... ... 4242.18 4242.14 B26 4990.40 4990.42 ... ... 5068.65 5068.52 B27 5791.98 5792.00 ... ... 5887.79 5887.64 Te(X-A) ... 4839.99 4803.18 ... ... 5406.00 5406.00 Te(A-B) ... 15 387.67 15 424.48 ... ... 15 283.00 15 282.00 aFor the 5-zeta and increased (9,4,4) active space models, only the ground state has been calculated. bMain model: MRCI calculation with aug-cc-pVQZ basis set (8,3,3) active space. cMain model plus MVD1 relativistic correction. dMain model plus 5-Z basis set and Douglas-Kroll relativistic correction. eMain model plus increased (9,4,4) active space. The ab initio EAMCs and SOCs between the electronic states are particularly important in the region around the crossings between the states in question. The angular momentum matrix elements have been calculated from the wavefunctions obtained after the MRCI cycle. The level of theory used is the same as for the potential energies. The spin-orbit (SO) matrix elements were calculated using the full Breit-Pauli operator as implemented in the MOLPRO package. The results of the spin-orbit and EAM calculations are illustrated in Table III, where some equilibrium SO values are compared to FIG. 1. Ab initio and refined potential energy curves (cm−1) for the first three electronic states of AlO. the values from the literature.39,42,43 The agreement is reasonable in all cases. A factor √2 must be applied to the value of HSO el (X−A) from Launila and Jonsson39 in order to agree with our definition of the off-diagonal SO constant X2+|ˆ HSO|A2. As a reference, the atomic limit of the spin-orbit coupling is 56.3 cm−1which is half energy splitting of ground state atomic aluminum 112.061 cm−1.43 III. ROVIBRONIC PROBLEM Although solution of the uncoupled diatomic nuclear motion problem is standard,44 there appears to be no general program available for solving the coupled problem. We have therefore developed a new program, DUO, to address this problem. DUO is a flexible, multistate program full details of which will be reported elsewhere.45 Here, we concentrate on aspects of the program important for AlO. Within the Born-Oppenheimer approximation, the SO free Hamiltonian of the diatomic problem in the absence of the hyperfine splitting is46 ˆ Htot =ˆ He+ˆ Hr+ˆ Hv,(1) where ˆ Heis the electronic part, ˆ Hris the rotational part, and ˆ Hvis the vibrational part. The rotational angular momentum ˆ Ris perpendicular to the internuclear axis. Considering Hund’s case (a) and representing the rotational
144312-4 Patrascu et al. J. Chem. Phys. 141, 144312 (2014) FIG. 2. Phase correction for the off-diagonal couplings and transition dipoles. See text for the definition of lB−A,bX−A,bB−A,andμX−A. angular momentum as ˆ R=ˆ J−ˆ L−ˆ S(2) in terms of the total ˆ J, electronic ˆ L, and spin ˆ Sangular momenta and choosing the body-fixed zaxis along the internuclear axis the rotational part is given by ˆ Hr= ¯2 2μR2ˆ J2−ˆ J2 z+ˆ L2−ˆ L2 z+ˆ S2−ˆ S2 z +(ˆ J+ˆ S−+ˆ J−ˆ S+)−(ˆ J+ˆ L−+ˆ J−ˆ L+) +(ˆ S+ˆ L−+ˆ S+ˆ L+),(3) where ˆ J±=ˆ Jx±iˆ Jy,ˆ S±=ˆ Sx±iˆ Sy, and ˆ L±=Lx±iLy are the corresponding angular momentum ladder operators and μ=mAmB/(mA+mB) is the reduced mass. The DUO basis set functions are chosen as the product |state,,S,,v=|state,,S,|J,,M|state,v, (4) where |state, ,S,is the electronic function, |J,,Mis the rotational function, |state,vis the vibrational function; “state” is the label identifying the electronic state; ,, and are the zaxis projections of the electronic, spin, and total angular momenta, respectively, and =+;Mis the FIG. 3. Ab initio and refined SOCs and EAMCs for the three lowest electronic states of AlO.
144312-5 Patrascu et al. J. Chem. Phys. 141, 144312 (2014) TABLE III. Ab initio and empirical values of SO and EAM matrix elements of 27Al16Oincm −1at R=1.77 Å. ab initio Refined Expt. Atomica SOA−A(cm−1)−61.10 −63.50 −63.38b−56.03 SOX−A(cm−1) 41.35 51.60 −45.66c SOB−A(cm−1) 48.63 62.78 LX−A(atomic units) −0.73 −0.51 −0.54d LB−A(atomic units) −1.17 −1.33 aObtained as minus half the atomic limit SO-splitting.43 bThe SO constant is related to the experimentally derived value Aeof the groundvibrational SO-splitting by Launila and Jonsson39 as −Ae/2; our conventions Eqs. (16) and (17) lead a sign change. cResult scaled by √2 with respect to Ref. 42. dA sign change due to our conventions Eqs. (13) and (14) is applied to the results of Ref. 42. projection of the total angular momentum along the laboratory axis Z;vis the vibrational quantum number. For the rotational basis set, the rigid rotor functions are used, with the following matrix elements: J,|ˆ J2|J,=J(J+1),(5) J,|ˆ Jz|J,=, (6) J, ∓1|ˆ J±|J,=J(J+1) −(∓1),(7) where Mis omitted for simplicity. The electronic basis functions appear in the solution only implicitly as matrix elements of different components of the operator Hras well as of the spin-orbit contributions. We choose these basis functions to satisfy the following conditions:46 state,,S,|ˆ Lz|state,,S,=, (8) σv(xz)|state,,S,=(−1)s−+S−|state,−, S, −, (9) where σv(xz) is a reflection through the molecular-fixed xzplane (parity operator) and s=1for|−states and s=0for all other states. The following non-vanishing matrix elements of the spin operators are valid: , S, |ˆ S2|, S, =S(S+1),(10) , S, |ˆ Sz|, S, =, (11) , S, ±1|ˆ S±|S,=S(S+1) −(±1),(12) where the label “states” is omitted for simplicity. The non-vanishing and symmetrically unique matrix elements of the ladder angular momentum operator ˆ L+as well as of the SO-matrix elements on these basis functions for the three lowest states of AlO X2+,A2, and B2+are obtained using MOLPRO and can be summarized as follows: LX−A=X2+,=0|ˆ L+|A2, =−1,(13) LB−A=B2+,=0|ˆ L+|A2, =−1,(14) SOA−A=A2, =1, =1 2ˆ HSOA2, (15) =1, =1 2, SOX−A=X2+,=0, =−1 2ˆ HSOA2, (16) =−1, =1 2, SOB−A=X2+,=0, =−1 2ˆ HSOA2, (17) =−1, =1 2, where S =1 2is omitted for simplicity. All other non-vanishing matrix elements can be obtained from Eqs. (13) to (17) by applying the symmetric properties upon the reflection as defined by Eq. (9) and the Wigner-Eckart theorem. In DUO, the electric dipole moment components μα (α=x,y,z) should be transformed into the standard irreducible representation as given by μ0=μz,(18) μ±=∓1 √2μx±iμx. In this representation, the diagonal and non-diagonal matrix elements of the (transition) dipole moments are μX−X=X2+,=0|μX−X 0|X2+,=0,(19) μA−A=A2, =1|μA−A 0|A2, =1,(20) μB−B=B2+,=0|μB−B 0|B2+,=0,(21) μX−A=X2+,=0|μX−A +|A2, =−1,(22) μB−A=B2+,=0|μB−A +|A2, =−1,(23) where values (=) are omitted for simplicity. It should be noted that for = 0 the MOLPRO functions are not eigenfunctions of ˆ Lzas required by Eq. (8).For example, the electronic state A2gives rise to two degenerate solutions labeled in MOLPRO as |1.2=|πxand |1.3 =|πy. The corresponding matrix representation of the ˆ Lz operator is not diagonal as computed by MOLPRO with the matrix elements being given, to within a global phase factor, by A2, πx|ˆ Lz|A2, πx=0,(24) A2, πy|ˆ Lz|A2, πy=0,(25) A2, πx|ˆ Lz|A2, πy=−i. (26) Besides the matrix elements of ˆ Lxand ˆ Lyas well as the spin-orbit matrix elements are all complex numbers in the MOLPRO representation. The transformation from |A2,πx,
144312-6 Patrascu et al. J. Chem. Phys. 141, 144312 (2014) S,and |A2,πy,S,(MOLPRO) to our representation |A2,=±1, S,(DUO) as defined by the conditions in Eqs. (8) and (9) is then given by |±= 1 √2[∓|x−i|y].(27) In this representation, the matrix representations of all couplings relevant to this work are real, including that for ˆ Lz which is diagonal. For example, the ˆ Lxand ˆ Lyoperators couple the Xand Astates via the following matrix elements in the MOLPRO representation: X2+|ˆ Lx|A2, πy=X2+|ˆ Ly|A2, πx≡ilX−A, (28) where lX−Ais a real number. Applying Eq. (27) we obtain LX−A=− √2lX−A.(29) Similarly, for the diagonal SO component coupling two components of the A2state, MOLPRO gives the following nonvanishing matrix elements: A2, πx, =1 2ˆ HSOA2, πy, =1 2=iaA−A,(30) A2, πx, =−1 2ˆ HSOA2, πy, =−1 2=−iaA−A, (31) where aA−Ais a real number. These couplings in the representation of Eq. (27) become real SOA−A=−aA−A.(32) The (non-vanishing) non-diagonal SO-matrix elements in the MOLPRO representation X2+, =−1 2ˆ HSOA2, πx, =1 2=bX−A, X2+, =−1 2ˆ HSOA2, πy, =1 2=ibX−A, X2+, =1 2ˆ HSOA2, πx, =−1 2=−bX−A, X2+, =1 2ˆ HSOA2, πy, =−1 2=ibX−A, B2+, =−1 2ˆ HSOA2, πx, =1 2=bB−A, B2+, =−1 2ˆ HSOA2, πy, =1 2=ibB−A, B2+, =1 2ˆ HSOA2, πx, =−1 2=−bB−A, B2+, =1 2ˆ HSOA2, πy, =−1 2=ibB−A, bX−Aand bB−Aare real numbers. In our representations, the non-vanishing and symmetrically unique values are SOX−A=√2bX−A,(33) SOB−A=√2bB−A.(34) The ˆ L2term in Eq. (3) depends only on the internuclear distance and thus can be included into the functional forms of the potential energy functions or indirectly into the SO coupling terms, and thus can be excluded from our analysis. The vibrational basis functions |state,vare prepared as solutions of the pure vibrational uncoupled eigen-problems for a given adiabatic electronic “state” − ¯2 2μ ∂2 ∂R2+Vstate(R)|state,v=Ev|state,v,(35) using an equidistant grid representation.47 Here, Vstate is the corresponding potential energy function. For the X2+, A2, and B2+electronic states, we select 60, 50, and 40 lowest eigenfunctions, respectively, as contracted basis functions for the rovibronic problem. The vibrational matrix elements for all R-dependent terms appearing in the rovibronic Hamiltonian are evaluated numerically using the rectangle rule. We used an equidistant grid of 2001 points ranging from 1.1 to 2.7 Å. Finally, the spin-rovibronic functions are symmetrized to be eigenfunctions of the parity operator σv(xz)usingEq.(9). Thus, for each total angular momentum quantum number J up to 112.5 and each of two parities +and −a Hamiltonian matrix is generated and then diagonalized using the LAPACK routine DSYEVR as provided by the intel MKL libraries. IV. REFINEMENT OF THE THEORETICAL RESULTS Our ab initio calculations are some distance from achieving spectroscopic accuracy. In order to approach this level of accuracy, we refine the potential energy curves and coupling terms using experimental data and employing the program DUO. The best spectroscopic data are available in the form of measured transition frequencies with an uncertainty of 0.023 cm−1by Saksena et al.23 or 0.019 cm−1by Launila and Berg25 (B−Xand A−Xsystem). In order to improve the quality of the rovibronic energies, the underlying ab initio energies were refined by fitting to the experimental data from Launila and Berg,25 as described below. The least square fit both to energy levels (term values) and transition frequencies is implemented in DUO. We started by fitting energies, which is technically simpler than fitting frequencies. Vibrational states up to v=7 in the case of X2+,uptov=16 in the case of A2, and v=6for B2+were included. We used experimental line positions from Launila and Berg25 and extracted the underlying energy levels using combination differences. A total of 5969 energy term values, with Jup to 112.5, reproduce the 23 793 frequencies with the root-mean-square (rms) error of 0.07 cm−1, with residuals varying up to 0.6 cm−1. The latter value is several times larger than the rms errors stated by either Saksena et al.23 or Launila and Berg,25 which might indicate some assignment problems. Indeed, we found more than 2000 duplicate frequencies having the same values but very different assignments. Furthermore, the experimental set due to Launila and Berg25 contains about 500 transitions that cannot be confirmed by combination differences. These transitions and the duplicates proved to be the largest outliers in our calculations. The PECs were fitted using the Extended Morse Oscillator (EMOs) functions44 given in the form V(R)=Te+De1−exp − N k=0 AkξkR−Re2 , (36)
144312-7 Patrascu et al. J. Chem. Phys. 141, 144312 (2014) TABLE IV. Parameters for refined potential energy curves given using an extended Morse oscillatory form, see Eq. (36); units are cm−1and Å. Parameter X2+A2+B2+ Te0.0 5405.85106434 20 688.31037841 Re=Rref 1.61890913 1.76905565 1.66807337 Te+De39 435.48544142 39 435.48544142 70 000.0 pl444 pr442 Nl222 Nr454 A01.90354673020 1.52647207263 1.51006043336 A1−0.10019383862 −0.16794482851 −0.14541739107 A2−0.07118881034 0.05855136709 0.04527686438 A3−0.23780463076 0.07637269923 1.28417838327 A40.27801043427 −0.14954775136 1.72071077730 A50.32678665277 where Reis an equilibrium distance and ξis the Šurkusvariable48 given by ξ=Rp−Rp ref Rp+Rp ref (37) with pas a parameter. This form guarantees the correct dissociation limit and allows for extra flexibility in the degree of the polynomial on the left or on the right sides of a reference position Rref whichwetakeatRref =re. This is specified by the parameters N=Nl(Nr) and p=pl(pr), respectively (see Table IV which gives the final fitted PEC parameters). The functional form of Eq. (36) is adapted when implemented on the right side of Rref =reby changing Nto Nrwhile, when used on the left side of Rref =reby changing Nto Nl.Alsothe power of the radius entering the Šurkus variable changes into pr, respectively, placcording to the region that is modeled. In the fits we used the same dissociation limit for the Xand Astates. The value that we determined is 39 435 cm−1(4.89 eV) which can be compared to the published experimental values for the ground state dissociation energy De=4.15 ±0.05 eV49 and 4.54 ±0.01 eV.50 The Bstate dissociation limit was constrained to lie 70 000 cm−1above the ground state potential minimum which is close to value given by considering the excitation of atomic oxygen. In order to describe the variation of the spin-orbit and electronic angular momentum couplings from the corresponding ab initio curves, we employed the morphing approach51,52 as follows. A refined function F(R) at a given grid point Ri(i=0...2001) is given by F(Ri)=H(Ri)Fai(Ri),(38) where Fai(Ri)isanab initio function obtained from a sparse grid of 105 geometries using the cubic splines interpolation. The morphing function H(R) is represented by a polynomial,44 H(R)=⎡ ⎣(1 −ξ) NT k≥0 Akξk+ξt∞⎤ ⎦,(39) in terms of the Šurkus variable ξ(Eq. (37)) with p=2 and the reference geometry chosen as Rref =1.77 Å. Here, TABLE V. Parameters of the refined morphing functions describing the SO and EAM functions of AlO (unitless), see Eqs. (38) and (39), as well of the empirical SR (cm−1) and BOB corrections (unitless). Object A0A1 SOA−A1.038991286985 −0.039342225874 SOX−A0.882363315197 SOB−A1.291104563209 LX−A0.699772845944 LB−A1.132563321574 SRX0.006749015759 SRA−0.002060781906 SRB−0.001688949114 BOBX0.001346178878 BOBA0.001183536766 BOBB0.001080261641 t∞=1 is the value of the morphing function at r→∞and the morphing expansion parameters are listed in Table V.For the interested reader, the supplementary material contains the final morphed curves on the fine grid. In order to target the experimental accuracy, we had to add two types of empirical corrections not available in MOLPRO. To account for the Born-OppenheimerBreakdown (BOB) effect,44 the rotational kinetic energy operator in Eq. (3) was replaced by ¯2 2μR2→ ¯2 2μR2(1 +gBOB(R)),(40) where the unitless BOB functions gBOB are represented by the expansion in Eq. (39) with state-specific expansion parameters. The resulting empirical expansion parameters are given in Table V. Finally, the spin-rotation (SR) couplings were also included as defined by SRX=X2+,=0,=1 2ˆ HSRX2+,=0,=1 2, (41) SRA=A2, =1, =1 2ˆ HSRA2, =1, =1 2, (42) SRB=B2+,=0,=1 2ˆ HSRB2+,=0, =1 2, (43) where the effective spin-rotational Hamiltonian term HSR is given by46 ˆ HSR =γ(R)( ˆ J−ˆ L−ˆ S)·ˆ S. (44) We also used Eq. (39) to represent the SR couplings γ(R) analytically. The expansion parameters obtained through the fits to the experimental data are listed in Table V. In the refinement up to 28 experimental energy levels per value of the quantum number Jfor even and odd parity states were used. The goal of this process is to obtain the best fit with the minimal number of parameters to ensure that the resulting model is capable of giving high accuracy predictions
144312-8 Patrascu et al. J. Chem. Phys. 141, 144312 (2014) FIG. 4. Residuals (observed-calculated, cm−1) for energies (upper display) and frequencies (lower display) of 27Al16O: The calculated values are obtained from our refined PECs, SOs, and EAMCs, while the observed are by Launila and Berg.25 that extend beyond the range of experimental data. The first step was to obtain the most accurate fit for low-lying energies which involved mainly the refinement of the lowest electronic state. In the fitting the shapes of refined curves are controlled and (partially) constrained to the corresponding original ab initio curves along the idea used successfully for polyatomic molecules.53–55 The program DUO allows the use of factors giving the relative weights to the ab initio or experimental data in a given fit. The fitting procedure started with a strong ab initio bias, but slowly increased the weight of the experimental fit up to a match with the weight for the ab initio fit. At this point, the overall shape of the curves was preserved but the accuracy was still not ideal. In order to improve on this, we continued to increase the weight for the experimental results. In this way, we obtained the equilibrium positions compatible with experiment, implicitly allowing flexibility in performing the refinement. When the lower energy term values were reproduced with a rms deviation better than 0.08 cm−1we added the higher electronic states, namely, the A2and B2+states. Optimization for the B2+state was straightforward. For the A2state, extra care was needed due to the fact that the spin-orbit coupling manifests itself by splitting the potential energy curve. This made the accurate fitting to experiment more difficult. Considering all the corrections and the refinement techniques used the energy levels obtained are in reasonable agreement with the observed ones up to the desired value of J =112.5 and our final fit reproduces our entire data set of energies with a rms error of 0.08 cm−1. At this stage, the experimental frequencies were added to the fit in order to exclude possible artefacts associated with the energy fit. The final rms error of the combined fit to both energies and frequencies is 0.07 cm−1, which is consistent with the error of our combination difference procedure (see above). The rms error of the frequencies only is 0.06 cm−1. A more detailed, graphical representation of the quality of the fit is shown in Fig 4, where Obs.−Calc. residuals for the energies and frequencies are presented. The fitting algorithms described above have been implemented as a standard part of DUO. Convergence of the refinement algorithm is fast, accurate fits being obtained in less than 10 iterations. The resulting numerical refined curves are in good agreement with the ab initio calculations for the range we are interested in, see Fig. 1. Figure 3also compares our ab initio and refined angular momentum coupling matrix element curves. These couplings are actually rather small and not too much significance should be given to the differences between the ab initio and refined values. The refinement of the coupling curves however deserves some attention: first, one can see from Fig. 3that the TABLE VI. Comparison of “observed”23 and calculated J=0.5 rovibronic energy levels for 27Al16Oincm −1. The electronic states are labelled X,A,andB for X2+,A2,andB2+, respectively. p=±is the total parity and =+. JpState v||Obs. Calc. Obs.−Calc. JpState v||Obs. Calc. Obs.−Calc. 0.5 +X0 0.5 0.0000 0.0000 0.0000 0.5 ... X0 0.5 1.2259 1.2742 −0.0483 0.5 +X1 0.5 965.4519 965.4355 0.0164 0.5 ... X1 0.5 966.6633 966.6994 −0.0361 0.5 +X2 0.5 1916.8596 1916.8454 0.0142 0.5 ... X2 0.5 1918.0589 1918.0993 −0.0404 0.5 +X3 0.5 2854.2366 2854.2063 0.0303 0.5 ... X3 0.5 2855.4091 2855.4508 −0.0417 0.5 +X4 0.5 3777.5016 3777.5042 −0.0026 0.5 ... X4 0.5 3778.6781 3778.7404 −0.0623 0.5 +X5 0.5 4686.7136 4686.6609 0.0527 0.5 ... X5 0.5 4687.8785 4687.8919 −0.0134 0.5 +A0 0.5 5346.1146 5346.1164 −0.0018 0.5 ... X6 0.5 5583.0853 5583.1308 −0.0455 0.5 +X6 0.5 5581.9326 5581.9076 0.0250 0.5 ... X7 0.5 6464.2647 6464.2586 0.0061 0.5 +A1 0.5 6067.0507 6066.9348 0.1159 0.5 ... B0 0.5 20 636.4057 20 636.4968 −0.0911 0.5 +X7 0.5 6463.1256 6463.0405 0.0851 0.5 ... B1 0.5 21 499.5010 21 499.4728 0.0282 0.5 +B0 0.5 20 635.3658 20 635.3089 0.0569 0.5 ... B2 0.5 22 355.2627 22 355.3500 −0.0873 0.5 +B1 0.5 21 498.4723 21 498.2939 0.1784 0.5 ... B3 0.5 23 203.6592 23 203.8278 −0.1686 0.5 +B2 0.5 22 354.2179 22 354.1801 0.0378 0.5 ... B4 0.5 24 044.7784 24 044.9011 −0.1227 0.5 +B3 0.5 23 202.6463 23 202.6668 −0.0205 0.5 ... B5 0.5 24 878.5385 24 878.5962 −0.0577 0.5 +B4 0.5 24 043.7747 24 043.7490 0.0257 0.5 ... B6 0.5 25 705.0220 25 704.9628 0.0592 0.5 +B5 0.5 24 877.5507 24 877.4529 0.0978 0.5 ... B7 0.5 26 524.1323 26 524.0736 0.0587 0.5 +B6 0.5 25 704.0201 25 703.8281 0.1920 0.5 ... B8 0.5 27 335.9454 27 336.0012 −0.0558 0.5 +B7 0.5 26 523.1582 26 522.9476 0.2106 0.5 +B8 0.5 27 334.9814 27 334.8839 0.0975