Multireference Configuration Interaction Wave Functions Research Articles

Calculation of transition moments between internally contracted MRCI wave functions with non-orthogonal orbitals†

We present an efficient algorithm for calculating transition properties between internally contracted multireference configuration interaction (MRCI) wave functions using different orbitals for both states. The method is applied to low lying electronic states of aluminium monoxide, and transition moment functions computed with the same or different orbital sets for the individual states are compared. The difference between transition moments obtained with state-averaged or state-specific orbitals is found to be rather small at the MRCI level, but substantial at the CASSCF level. †Dedicated to Professor Pavel Rosmus.

Seams near seams: The Jahn-Teller effect in the E″1 state of N3+

The (1)E(") electronic state of cyclic N(3) (+) arising from the singly excited electron configuration e('3)e(") is studied using multireference configuration interaction wave functions and a quadratic Jahn-Teller Hamiltonian determined from those calculations. It is shown that these two states have both a symmetry-required seam of conical intersections at D(3h) geometries and three proximal symmetry equivalent C(2v) seams, located on a circle with radius rho(0) from the D(3h) intersection. rho(0), a function of Q(s), the breathing mode, is quite small but only attains a value of zero at Q(s) (crit)=1.252 A, resulting in a confluence or intersection node of the three C(2v) seams with the D(3h) seam. At this point only, g= parallelg(Q(s)) parallel, the norm of half the energy difference gradient, the linear Jahn-Teller term, vanishes and the intersection is of the Renner-Teller type. The close proximity of the previously unreported C(2v) seams to the D(3h) seam over the range of Q(s) considered is a consequence of the small values of g, compared to the quadratic Jahn-Teller term. The present analysis has important implications in the study of Jahn-Teller effects in ring systems and provides insight into a recent report that characterized this D(3h) seam as a Renner-Teller or glancing intersection.

Reexamination of the Lyman-Birge-Hopfield transition ofN2: Configuration-interaction generalized oscillator strengths

Generalized oscillator strengths (GOS) for the dipole forbidden quadrupole allowed Lyman-Birge-Hopfield (LBH) valence transition of ${\mathrm{N}}_{2}$ have been calculated. Complete active space self-consistent-field (CAS) and multireference configuration interaction (MRCI) wave functions of several types have been used and their effects on the computed GOS profiles discussed. Orthogonal and nonorthogonal molecular orbitals have been employed. The latter allows the explicit inclusion of relaxation effects in the excited state leading to a clear spectra interpretation. Hartree-Fock (HF) and CAS wave functions have been compared as a previous step before the MRCI calculations. Differences between the respective GOS profiles were found, with the CAS MRCI GOS curve being slightly closer to experiment than the HF MRCI one. Agreement between the present theoretical CAS MRCI GOS profile employing nonorthogonal orbitals and recent experimental results are very favorable, being the best one among available theoretical calculations in the literature.

Towards a highly efficient theoretical treatment of Jahn-Teller effects in molecular spectra: The 1A2 and 2A2 electronic states of the ethoxy radical

Nonadiabatic effects in the two lowest electronic states of the ethoxy radical, the 1 (2)A and 2 (2)A states, are considered, using multireference configuration interaction (MRCI) wave functions comprised of over 15x10(6) configuration state functions. The lowest point on the seam of conical intersection is located. Using this point as the origin, a quasidiabatic Hamiltonian suitable for use in a multimode vibronic coupling treatment of the coupled 1 (2)A and 2 (2)A electronic states is determined. The Hamiltonian includes all contributions from all internal coordinates through second order in displacements from the origin and is comprised of over 500 parameters. By using the average energy gradient, the energy difference gradients, and the derivative couplings, all of which are obtained at little additional cost once the requisite eigenstates are known, the second order Hamiltonian is determined from MRCI calculations at only 35 nuclear configurations. This is essentially the same number of points required to obtain the frequencies for the ground state equilibrium structure using centered differences of gradients. The diabatic Hamiltonian provides a good description of the seam space, the (N(int)-2)-dimensional space of conical intersection points, continuously connected to the minimum energy crossing point, enabling, for the first time, an analysis of the changes in the branching plane induced by seam curvature in the full seam space. Comparing the diabatic representation and MRCI results we find a good agreement for the ground state equilibrium structure, R(eq)(1 (2)A), as well as the ground state energy and vertical excitation energy. In good agreement with the available experimental data are the ground state equilibrium structure and the excitation energy to the A (2)A state, predicted here to involve a cone state level. Agreement between the harmonic frequencies at R(eq)(1 (2)A) computed from the MRCI wave function and from the diabatic Hamiltonian is excellent for all but the three lowest energy normal modes where significant deviations are observed indicating the need for selected cubic and/or quartic terms. For the low-lying vibrational levels, the diabatic representation can be used to partition the normal modes into two groups, those that involve inter(diabatic) state coupling and those that are spectators as far as nonadiabatic effects are concerned. The spin-orbit coupling interaction is determined using the Breit-Pauli approximation and its incorporation into the diabatic Hamiltonian is discussed.

An ab initio study of the low-lying electronic states of S3

Accurate calculations of the low-lying singlet and triplet electronic states of thiozone, S(3), have been carried out using large multireference configuration interaction wave functions. Cuts of the full potential energy surfaces along the stretching and bending coordinates have been presented, together with the vertical excitation spectra. The strong experimentally observed absorption around 395 nm is assigned to the 1 (1)B(2) state, which correlates to ground state products. Absorption at wavelengths shorter than 260 nm is predicted to lead to singlet excited state products, S(2) (a (1)Delta(g))+S((1)D). The spectroscopic properties of the X (3)Sigma(g) (-), a (1)Delta(g), and b (1)Sigma(g) (+) electronic states of the S(2) radical have also been accurately characterized in this work. The investigations of the low-lying electronic states were accompanied by accurate ground state coupled cluster calculations of the thermochemistry of both S(2) and S(3) using large correlation consistent basis sets with corrections for core-valence correlation, scalar relativity, and atomic spin-orbit effects. Resulting values for D(0)(S(2)+S) and SigmaD(0) for S(3) are predicted to be 61.3 and 162.7 kcal/mol, respectively, with conservative uncertainties of +/-1 kcal/mol. Analogous calculations predict the C(2v)-D(3h) (open-cyclic) isomerization energy of S(3) to be 4.4+/-0.5 kcal/mol.

Configuration interaction oscillator strengths of the H2O molecule: Transitions from the ground to the [formula omitted] and 11B2 excited states

The vertical optical oscillator strength (OOS) at the length (fL) and velocity (fV) forms for the {B∼1A1,C∼1B1,D∼1A1,11B2}←X∼1A1 transitions of H2O were computed employing complete active space self consistent field followed by multireference configuration interaction (CAS/MR-CI) wave functions and d-aug-cc-pVXZ (X=D,T,Q,5) Gaussian basis sets. The computed OOS values at the highest level, namely, CAS-MRCI/d-aug-cc-pV5Z, were (fL value, fV value): B∼1A1 (0.05494, 0.05993), C∼1B1 (0.008700, 0.009634), D∼1A1 (0.04695, 0.04880) and 11B2 (0.01740, 0.01905). These results are compared with previous theoretical and experimental values, with the latter when possible.

Computed lifetimes of metastable states of CO2+

Highly correlated internally contracted multireference configuration interaction wave functions are used to calculate the potential energy and spin-orbit coupling functions for the lowest electronic states of CO2+ dication. Using these functions, the positions and lifetimes of the corresponding vibronic states are evaluated by means of log-phase-amplitude, stabilization, and complex-scaling methods within the framework of a multichannel Schrodinger analysis. For the first time in the literature, the calculated lifetimes are in good agreement with the experiment, thereby proving the reliability of the predicted characteristics and adequacy of the used theory for a theoretical study of other molecular dications.

Reduced multireference coupled cluster method with singles and doubles: Perturbative corrections for triples

The reduced multireference coupled-cluster method with singles and doubles (RMR CCSD) that employs multireference configuration interaction wave function as an external source for a small subset of approximate connected triples and quadruples, is perturbatively corrected for the remaining triples along the same lines as in the standard CCSD(T) method. The performance of the resulting RMR CCSD(T) method is tested on four molecular systems, namely, the HF and F(2) molecules, the NO radical, and the F(2) (+) cation, representing distinct types of molecular structure, using up to and including a cc-pVQZ basis set. The results are compared with those obtained with the standard CCSD(T), UCCSD(T), CCSD(2), and CR CCSD(T) methods, wherever applicable or available. An emphasis is made on the quality of the computed potentials in a broad range of internuclear separations and on the computed equilibrium spectroscopic properties, in particular, harmonic frequencies omega(e). It is shown that RMR CCSD(T) outperforms other triply corrected methods and is widely applicable.

Influences on the calculation of accurate and basis set extrapolated oscillator strengths: the transition of H2O

The vertical optical oscillator strength (OOS) for the transition was calculated using complete active space self-consistent field followed by multireference configuration interaction (CAS/MR-CI) wavefunctions and Gaussian basis sets of the aug-cc-pVXZ and d-aug-cc-pVXZ types (X = D, T, Q, 5). We discuss the dependence of the results on the size of the CAS space, the MR-CI calculation and the Gaussian basis set. The results were extrapolated using the F(X) = A + Bexp(−CX) function. The oscillator strengths were obtained at the length (fL) and velocity (fV) forms. The most extensive treatment gave fL and fV values, with extrapolated values between brackets, 0.048 74 (0.048 71) and 0.054 15 (0.054 12), respectively.

Electron spin resonance g tensors for complexes of Ne and Ar with AlO: Theoretical studies related to the large matrix effect observed for AlO

For Ne(n)-AlO (n=2, 4, 6, 8, 10) and Ar(n)-AlO clusters (n=2, 4, 6, 8), the perpendicular (relative to AlO) component of the g tensor was calculated by second-order perturbation theory, using multireference configuration-interaction wave functions. The rare-gas (Rg) atoms were placed axially and/or off axially (one or two rings of four Rg atoms each), and the distance of the Rg atoms from the Al and O atoms, or from the AlO axis, was varied from 4 to 12 bohrs. Rg atoms placed axially mostly increase g(perpendicular), whereas off-axially placed ones lower it below the gas-phase value of AlO. The largest deviations from g(perpendicular) of isolated AlO occur at Ne-Al,O distances of 5-6 bohrs, and Ar-Al,O distances of 6-9 bohrs, with maximal lowerings of about 1600 ppm for Ne and about 2200 ppm (estimated) for Ar in the case of two axial and eight off-axial Rg atoms. Electron spin resonance studies by Knight and Weltner found large matrix effects for AlO, with downshifts of g(perpendicular) observed to be about 450 and 1150 ppm in Ne and Ar matrices, respectively.

Theoretical study of the photodetachment spectrum of the methylene amidogene anion (H2CN−)

The excited states of methylene amidogen (H2CN) have recently been studied extensively both by experiment and theory. However, the assignment of absorption spectra is in disagreement with the theoretical analysis. In the present study a theoretical photoelectron detachment spectrum is predicted which shows that a corresponding experiment could be suitable to decide whether the theoretical or the experimental assignment is correct. To this end, the electron affinity of H2CN was calculated very accurately. Vertical and adiabatic electron detachment energies were calculated by complete active space and multi-configuration self-consistent field (CASSCF and MCSCF) as well as by multireference configuration interaction (MRCI) methods. The MRCI wave functions were utilized to compute electronic detachment probabilities. Beyond the radical ground state, three significant detachments are found in the region around 4-5 eV and a further weak detachment at roughly 8.4 eV. Vibrational intensities were determined for the four lowest electronic states of the radical and vibrationally resolved photodetachment spectra were constructed. The predicted spectra show that the bands of the first 2B1 and 2A1 states are clearly distiguishable while the band for the nearby second 2B1 state is weak and strongly overlapping with the 2A1 manifold.

The [formula omitted] and [formula omitted] states of N 2+: hyperfine and nuclear quadrupole coupling constants, electric quadrupole moments, and electron-spin g-factors. A theoretical study

The variation with bond distance of hyperfine coupling constants (hfcc) [ A iso, A dip], nuclear quadrupole coupling constants (nqcc) [ χ], molecular electric quadrupole moments [ Θ], and electron-spin g-factors is reported for X 2 Σ g +(1σ u 21π u 42σ g) and B 2 Σ u +(1σ u1π u 42σ g 2) of N 2 +. The hfcc’s, nqcc’s, and Θ’s are studied with ab initio and density functional theory methods. Calculated A iso/ A dip of 91.3/29.7 MHz for the X state have to be compared with experimental values of 102.4/23.3 MHz. For the B state, theoretical values A iso/ A dip=793/6.8 MHz support an experimental A iso=708 MHz (1 σ u is largely s) but disagree with A dip=26 MHz (large p). The nqcc χ( B) is one order of magnitude larger than χ( X), 4.80 versus −0.58 MHz, in line with literature results. Omission of nqcc contributions in the analysis of optical spectra is most probably responsible for the anomalously high “experimental” A dip( B). The quadrupole moment Θ is positive for both X and B states of N 2 +, whereas it is negative for neutral N 2( X). The vibrational absorption intensity for the v=1←0 quadrupole transition of N 2 +( X) is about six times stronger than for N 2( X). The g ⊥-shift (Δ g ⊥= g ⊥− g e ) is evaluated with multireference configuration interaction wavefunctions, via second-order sum-over-states expansions. The Δ g ∥’s are small (≈−100 ppm), while Δ g ⊥( v=0) is −2710 ppm for X 2 Σ g + and −6165 ppm for B 2 Σ u + . Interaction parameters | α el|=31.12 cm −1 and β el=1.01 a.u. calculated for the A 2 Π u∼B 2 Σ u + perturbation are in line with 31.30 cm −1 and 1.07 a.u. averaged over five experimental studies. Our Δ g ⊥( v)’s are compared with Curl’s shifts obtained from experimental spin-rotation ( γ) and rotational constants. Good agreement is found for X but not for the B state, for which it is suggested that experimental γ( B, v)’s recently reported—where γ decreases sharply with vibrational level v—should be revised.

Analytic evaluation of nonadiabatic coupling terms at the MR-CI level. I. Formalism.

An efficient and general method for the analytic computation of the nonandiabatic coupling vector at the multireference configuration interaction (MR-CI) level is presented. This method is based on a previously developed formalism for analytic MR-CI gradients adapted to the use for the computation of nonadiabatic coupling terms. As was the case for the analytic energy gradients, very general, separate choices of invariant orbital subspaces at the multiconfiguration self-consistent field and MR-CI levels are possible, allowing flexible selections of MR-CI wave functions. The computational cost for the calculation of the nonadiabatic coupling vector at the MR-CI level is far below the cost for the energy calculation. In this paper the formalism of the method is presented and in the following paper [Dallos et al., J. Chem. Phys. 120, 7330 (2004)] applications concerning the optimization of minima on the crossing seam are described.

Global ab initio potential energy surfaces for the lowest three doublet states (1 2A′, 2 2A′, and 1 2A″) of the BrH2 system

Global adiabatic potential energy surfaces (PESs) of the lowest three doublet states (1 A2A′, 2 2A′, and 1 2A″) for the BrH2 system have been calculated using the multireference configuration interaction (MRCI) method including the Davidson’s correction (Q) with the aug-cc-pVTZ basis set. Spin–orbit effects were considered on the basis of the Breit–Pauli Hamiltonian using the MRCI wave functions. The calculated adiabatic energies were fitted to the analytical functional form of many-body expansion. The barrier heights of the H+HBr→H2+Br abstraction and H+H′Br→H′+HBr exchange reactions on the ground-state PES were calculated to be 1.28 and 11.71 kcal mol−1, respectively, both of which are slightly smaller than the values obtained in the previous work [G. C. Lynch, D. G. Truhlar, F. B. Brown, and J.-G. Zhao, J. Phys. Chem. 99, 207 (1995)]. The fits for the 1 2A′, 2 2A′, and 1 2A″ PESs were successful within an accuracy of 0.1 kcal mol−1 in the important regions of PESs such as the transition states and van der Waals wells. Thermal rate constants for the abstraction and exchange reactions and their isotopic variants were calculated with the fitted 1 2A′ PES using the improved canonical variational transition-state theory with the least-action adiabatic ground-state approximation method. The calculated rate constants were found to agree better with experiment than those obtained by Lynch et al.

Geometries, Electronic g-tensor Elements, Hyperfine Coupling Constants, and Vertical Excitation Energies for Small Gallium Arsenide Doublet Radicals, GaxAsy (x + y = 3, 5)

Geometries of the gallium arsenide doublet radicals GaAs 2 , Ga 2 As, Ga 2 As 3 , Ga 3 As 2 , GaAs 4 , and Ga 4 As were optimized by the B3LYP/6-311+G(2df) method and compared with literature values. For the global minimum, as well as for isomers lying up to 0.2 eV higher, hyperfine coupling constants (HFCC) and electron-spin g-tensors were calculated. For HFCCs the B3LYP/6-311+G(2df) method was used, whereas for g-tensors second-order perturbation calculations with multireference configuration interaction wave functions and a valence triple-< basis set with polarization functions (TZVP) were performed. Generally, due to the low s-spin and high p-spin densities, A i s o values are small, and A d i p 's large. The g-shifts (Δg = g - g e ) are on the order of 100 000 ppm, caused by large spin-orbit couplings and low excitation energies. For the experimentally known Ga 2 As 3 , values calculated for the D 3 h structure are (A's in MHz, Δg's in ppm) A i s o ( 6 9 Ga) = 1325 (1524); A i s o ( 7 5 As) = -23 (65); A d i p ( 6 9 Ga) = 65 (87); A d i p ( 7 5 As) = 36 (0); Δg⊥ = -73 410 (-82 300); and Δg∥ = 6460 (0), with magnetic parameters derived from the experimental values in parentheses. Mulliken spin densities are shown to be a good measure of A d i p values. Vertical excitation energies, as obtained from the g-tensor calculations, are also tabulated.

Size-consistent multireference configuration interaction method through the dressing of the norm of determinants

A new determinant-specific, effective change (‘dressing’) of the norm of the multireference configuration interaction (MRCI) wavefunction is proposed in order to achieve the size-consistency of the MRCI method. The new approach provides a unifying framework for analysis of size-consistent extensions of the MRCI method that are based on the coupled pair functional (CPF) strategy and lead to simplified computations of the analytical gradients. Using the new framework, a generalized multireference full coupled pair functional (MR-FCPF) method is introduced. The MR-FCPF method may be viewed as a functional counterpart of the recently proposed generalized (‘full’) coupled electron pair approximation (CEPA), referred to as the size-consistent self-consistent CI ((SC)2CI) method. A straightforward extension of the MR-FCPF method leads to a pseudo-functional form of the coupled cluster (CC) type formalisms. Therefore, the new approach may be used to introduce a simple alternative to existing CC-type gradient techniques. The new procedure is formally derived and compared with similar methods from the literature. Model systems calculations (H2O, LiF, CH+ 2) are further used to demonstrate the effect of various approximations and to elucidate the hierarchy of functional MR-CEPA schemes.

Improved calculation of spin-dependent properties for internally contracted multireference configuration interaction wavefunctions

A general approach to evaluate spin-dependent properties for internally contracted multireference configuration interaction wavefunctions is presented. The spin-orbit atomic integral program as implemented within MOLPRO is extended to treat arbitrary basis sets, and the spin-spin atomic integrals are added. The increased accuracy of the code is demonstrated by means of calculations of the fine structures of the lowest levels of O and O+.

First principles prediction of isotopic shifts in H2O

We compute isotope-independent first- and second-order corrections to the Born–Oppenheimer approximation for water and use them to predict isotopic shifts. For the diagonal correction, we use internally contracted multireference configuration interaction wave functions and derivatives with respect to mass-dependent internal coordinates to generate the mass-independent correction functions. For the nonadiabatic correction, we use a self-consistent field wave function for the ground electronic state and single excitation configuration interaction wave functions for the excited states and a generalization of the Handy, Yamaguchi, and Schaefer method to obtain mass-independent correction functions. We find that including the nonadiabatic correction gives significantly improved results compared to just including the diagonal correction when the Born–Oppenheimer potential energy surface is optimized for H2 O16. The agreement with experimental results for deuterium- and tritium-containing isotopes is nearly as good as our best empirical correction, however, the present correction is expected to be more reliable for higher, uncharacterized, levels.

An accurate determination of rovibrational spectra using the externally corrected coupled-cluster approaches: LiH ground state

Recently acquired highly precise spectroscopic data for the ground state of LiH and its various isotopomers are employed to carry out a critical assessment of the performance of the externally corrected coupled-cluster (CC) approaches. Both the amplitude and energy-corrected approaches are considered, in particular the reduced multireference CC method with singles and doubles (RMR CCSD) and the asymmetric energy formula based CCSD-[MR] method, both exploiting the same modest-size multireference configuration interaction (MR CISD) wave function, based on an M-dimensional reference space, as the source of higher than pair clusters. To assess the size of the basis set errors relative to those of the methods employed, the comparison is also made with the full CI (FCI) results at the cc-pVTZ level. The rovibrational energy levels and the corresponding transition frequencies are then computed for various isotopomers of LiH using the theoretically determined potentials at the cc-pVXZ (X=D, T, Q, and 5) and the extrapolated completebasis-set-limit levels and a comparison is made with the experiment as well as with the existing theoretical results, particularly those exploiting the CC approaches. The role of adiabatic corrections is also assessed. It is shown that both the RMR CCSD and CCSD-[4R] methods, yielding an almost indistinguishable results, are capable of accounting for the nondynamical correlation effects that are lacking in the standard single-reference CCSD approaches.

The Rovibrational Dependence of the 14N Nuclear Quadrupole Coupling Constants in the X2Σ+ and B2Σ+ States of CN from the Multireference CI Approach

In using several augmented correlation-consistent basis sets and reference configuration spaces, the 14N quadrupole coupling constants (QCCs) of rovibrational levels of the X2Σ+ and B2Σ+ states of the CN radical are computed from internally contracted multireference configuration interaction wave functions. To examine the overall quality of the correlated wave functions used for computing the expectation values of the electric field gradient (EFG) tensor at the N nucleus, electric dipole moments are calculated and spectroscopic constants are derived from corresponding potential energy curves. The adequacy of the expectation value approach to the evaluation of the EFG and dipole moment is discussed. The calculated vibrational dependence of the 14N QCC compares reasonably with the available experimental data.