Generalized Van Vleck Perturbation Theory Research Articles

Potential-Energy Curves for the Ground and Several Electronic States of NdO and NdS.

Potential energy curves (PECs) were calculated for 21 and 18 electronic states of NdO and NdS molecules, respectively. In each case, static electron correlation effects were described by incomplete model space multiconfiguration self-consistent field wave functions based on an active space that included the most important valence orbitals. Dynamic electron correlation was included by the multireference second-order generalized Van Vleck perturbation theory method. Scalar-relativistic contributions were included by the effective core potential approach, using def2-TZVPP basis sets. Spin-dependent relativistic corrections were determined to be small and negligible for the Nd atom and so were not included in the calculations. The 21 and 18 electronic states of NdO and NdS were predicted to be in the excitation energy range of ∼3.2 and ∼2.7 eV, respectively. The ground electronic states of NdO and NdS were determined as 15H (6s4fσ4fϕ4fδ) and 15H (4fϕ4fπ4fπ6s), with spectroscopic constants: bond length Re = 1.780 and 2.325 Å, and harmonic frequency ωe = 891 and 538 cm-1, respectively.

Riemannian Trust Region Method for Minimization of the Fourth Central Moment for Localized Molecular Orbitals.

The importance of localized molecular orbitals (MOs) in correlation treatments beyond mean-field calculation and in the illustration of chemical bonding (and antibonding) can hardly be overstated. However, the generation of orthonormal localized occupied MOs is significantly more straightforward than obtaining orthonormal localized virtual MOs. Orthonormal MOs allow facile use of highly efficient group theoretical methods (e.g., graphical unitary group approach) for calculation of Hamiltonian matrix elements in multireference configuration interaction calculations (such as MRCISD) and in quasi-degenerate perturbation treatments, such as the Generalized Van Vleck Perturbation Theory. Moreover, localized MOs can elucidate qualitative understanding of bonding in molecules, in addition to high-accuracy quantitative descriptions. We adopt the powers of the fourth moment cost function introduced by Jørgensen and coworkers. Because the fourth moment cost functions are prone to having multiple negative Hessian eigenvalues when starting from easily available canonical (or near-canonical) MOs, standard optimization algorithms can fail to obtain the orbitals of the virtual or partially occupied spaces. To overcome this drawback, we applied a trust region algorithm on an orthonormal Riemannian manifold with an approximate retraction from the tangent space built into the first and second derivatives of the cost function. Moreover, the Riemannian trust region outer iterations were coupled to truncated Conjugate Gradient inner loops, which avoided any costly solutions of simultaneous linear equations or eigenvector/eigenvalue solutions. Numerical examples are provided on model systems, including the high-connectivity H10 set in 1-, 2-, and 3-dimensional arrangements, and on a chemically realistic description of cyclobutadiene (c-C4H4) and the propargyl radical (C3H3). In addition to demonstrating the algorithm on occupied and virtual blocks of orbitals, the method is also shown to work on the active space at the MCSCF level of theory.

Analytical double-unitary-transformation approach for strongly and periodically driven three-level systems

Floquet theory combined with the generalized Van Vleck nearly degenerate perturbation theory has been widely employed for studying various two-level systems that are driven by external fields via time-dependent longitudinal (i.e., diagonal) couplings. However, three-level systems strongly driven by time-dependent transverse (i.e., off-diagonal) couplings have rarely been investigated, due to the breakdown of the traditional rotating wave approximation. Meanwhile, the conventional perturbation theory is not directly applicable, since a small parameter for the perturbed part is no longer apparent. Here we develop a double-unitary-transformation approach to deal with periodically modulated and strongly driven systems, where the time-dependent Hamiltonian has large off-diagonal elements. The first unitary transformation converts strong off-diagonal elements to diagonal ones, and the second enables us to harness the generalized Van Vleck perturbation theory to deal with the transformed Floquet matrix and also allows us to reduce the infinite-dimensional Floquet Hamiltonian to a finite effective one. For a strongly modulated three-level system, with the combination of the Floquet theory and the transformed generalized Van Vleck perturbation theory, we obtain analytical results of the system, which agree well with exact numerical solutions. This method offers a useful tool to analytically study multilevel systems with strong transverse couplings.

Use of density functional theory orbitals in the GVVPT2 variant of second-order multistate multireference perturbation theory.

A new variation of the second-order generalized van Vleck perturbation theory (GVVPT2) for molecular electronic structure is suggested. In contrast to the established procedure, in which CASSCF or MCSCF orbitals are first obtained and subsequently used to define a many-electron model (or reference) space, the use of an orbital space obtained from the local density approximation (LDA) variant of density functional theory is considered. Through a final, noniterative diagonalization of an average Fock matrix within orbital subspaces, quasicanonical orbitals that are otherwise indistinguishable from quasicanonical orbitals obtained from a CASSCF or MCSCF calculation are obtained. Consequently, all advantages of the GVVPT2 method are retained, including use of macroconfigurations to define incomplete active spaces and rigorous avoidance of intruder states. The suggested variant is vetted on three well-known model problems: the symmetric stretching of the O-H bonds in water, the dissociation of N2, and the stretching of ground and excited states C2 to more than twice the equilibrium bond length of the ground state. It is observed that the LDA-based GVVPT2 calculations yield good results, of comparable quality to conventional CASSCF-based calculations. This is true even for the C2 model problem, in which the orbital space for each state was defined by the LDA orbitals. These results suggest that GVVPT2 can be applied to much larger problems than previously accessible.

GVVPT2 Multireference Perturbation Theory Description of Diatomic Scandium, Chromium, and Manganese

With relatively simple model spaces derived from valence bond models, a straightforward zero-order Hamiltonian, and the use of moderate-sized Dunning-type correlation consistent basis sets (cc-pVTZ, aug-cc-pVTZ, and cc-pVQZ), the second order generalized Van Vleck perturbation theory (GVVPT2) method is shown to produce potential energy curves (PECs) and spectroscopic constants close to experimental results for both ground and low-lying excited electronic states of Sc(2), Cr(2) and Mn(2). In spite of multiple quasidegeneracies (particularly for the cases of Sc(2) and Mn(2)), the GVVPT2 PECs are smooth with no discontinuities. Since these molecules have been identified as ones that widely used perturbative methods are inadequate for describing well, due to intruder state problems, unless shift parameters are introduced that can obfuscate the physics, this study suggests that the conclusion about the inadequacy of multireference perturbation theory be re-evaluated. The ground state of Sc(2) is predicted to be X(5)∑(u)(-), and its spectroscopic constants are close to the ones at the MRCISD level. Near equilibrium geometries, the 1(3)∑(u)(-) electronic state of Sc(2) is found to be less stable than the quintet ground state by 0.23 eV. The Cr(2) PEC has several features of the Rydberg-Klein-Rees (RKR) experimental curve (e.g., the pronounced shelf at elongated bond lengths), although the predicted bond length is slightly long (R(e) = 1.80 Å with cc-pVQZ compared to the experimental value of 1.68 Å). The X(1)∑(g)(+) ground state of Mn(2) is predicted to be a van der Waals molecule with a long bond length, R(e), of 3.83 Å using a cc-pVQZ basis set (experimental value = 3.40 Å) and a binding energy, D(e), of only 0.05 eV (experimental value = 0.1 eV). We obtained R(e) = 3.40 Å and D(e) = 0.09 eV at the complete basis set (CBS) limit for ground state Mn(2). Low lying excited state curves have also been characterized for all three cases (Cr(2), Mn(2), and Sc(2)) and show similar mathematical robustness as the ground states. These results suggest that the GVVPT2 multireference perturbation theory method is more broadly applicable than previously documented.

Nonadiabatic coupling terms for the GVVPT2 variant of multireference perturbation theory

A Lagrangian based approach was used to obtain analytic formulas for nonadiabatic coupling terms within a multireference perturbation theory description of molecular electronic structure. Specifically, formulas were developed for the second-order generalized Van Vleck perturbation theory (GVVPT2) description of electron correlation. The formalism can use either complete or incomplete model (or reference) spaces, and is limited, in this regard, only by the capabilities of the MCSCF program. Of particular interest, the suggested formalism can straightforwardly use state-averaged MCSCF descriptions of the reference space in which the states have arbitrary weights. Since GVVPT2 wave functions are not strictly orthogonal, the definition of nonadiabatic coupling terms is untrivial and a perturbation-order consistent definition is suggested herein from which working equations are developed.

Multireference composite approaches for the accurate study of ground and excited electronic states: C2, N2, and O2

A multireference analog of the correlation consistent composite approach (MR-ccCA) based on complete active space with second-order perturbation theory (CASPT2) has been utilized in an investigation of the ground and valence excited states of C(2), N(2), and O(2). The performance of different second-order multireference perturbation theory methods including second-order n-electron valence state perturbation theory, second-order multireference Møller-Plesset, and second-order generalized van Vleck perturbation theory has been analyzed as potential alternatives to CASPT2 within MR-ccCA. The MR-ccCA-P predicts spectroscopic constants with overall mean absolute deviations from experimental values of 0.0006 Å, 7.0 cm(-1), and 143 cm(-1) for equilibrium bond length (r(e)), harmonic frequency (ω(e)), and term values (T(e)), respectively, which are comparable to the predictions by more computationally costly multireference configuration interaction-based methods.

Multireference Generalized Van Vleck Perturbation Theory (GVVPT2) Study of the NCO + HCNO Reaction: Insight into Intermediates†

Possible mechanisms for the NCO + HCNO reaction, previously studied using single reference methods, have been revisited using the second-order generalized Van Vleck perturbation theory (GVVPT2) variant of multireference perturbation theory. Large scale GVVPT2 calculations, with 25 active orbitals, demonstrate that the electronic structures of the ground states of all eight previously identified plausible doublet spin intermediates of this reaction are substantially multiconfigurational with maximum configuration amplitudes not larger than 0.87. Moreover, comparison of GVVPT2 results with underlying MCSCF calculations demonstrates a pronounced effect of dynamic correlation on the configurational structure. GVVPT2 calculations of vertical excitation energies of the intermediates showed that all the intermediates, except one, have significant excitation energies to both the same and other spatial symmetries as the ground states. The relative GVVPT2 energies of the intermediates were found to be consistent with earlier calculations, including the prediction of a negative temperature dependence of the rate constant, and support available experimental observations.

Ground and Low-Lying Excited Electronic States of [3,3′] Bidiazirinylidene (C2N4)

The second-order generalized Van Vleck perturbation theory (GVVPT2) variant of multireference perturbation theory was used to investigate the ground and low-lying excited electronic states of [3,3'] bidiazirinylidene (C(2)N(4)). C(2)N(4) is arguably the simplest molecule with multiple CN(2) rings. Equilibrium geometries and adiabatic energy differences of the 1(1)A(g) ground state and 2 (1)A(g), 1(1)B(1g), 1(3)B(1g), 1(1)B(2g), 1(3)B(2g), 1(1)B(3g), 1(3)B(3g), 1(1)B(2u), 1(3)B(2u), 1(1)B(3u), and 1(3)B(3u) excited states of D(2h) symmetry were obtained. In addition, vertical excitation energies to and emission energies from the excited states were obtained. The 1(1)A(g) optimized geometry and harmonic frequencies for the CN(2) rings in C(2)N(4) are similar to those for the related difluorodiazirine molecule: R(CN), R(NN) bonds and the angleNCN angle are 0.012 and 0.006 A longer and 0.8 degrees smaller than that for the ring in F(2)CN(2). The NN stretch frequencies are 1470 and 1562 cm(-1), and CN stretches are 993 and 995 cm(-1). The similarity between the GVVPT2 results of C(2)N(4) and F(2)CN(2) suggests that C(2)N(4) might be sufficiently stable to be synthesized. The lowest-energy spin- and electric dipole-allowed vertical excitation, to the 1(1)B(2u) state, is predicted to occur in the visible (2.30 eV, 538.9 nm).

Molecular electric dipole moments using the GVVPT2 variant of multireference perturbation theory

A Lagrangian approach for electric dipoles within second order Generalized Van Vleck Perturbation Theory (GVVPT2) is presented. The Lagrangian approach for this response property requires that only one, perturbation-independent, set of simultaneous equations needs to be solved. The presented formulation is demonstrated numerically on several well studied molecules at their equilibrium bond lengths and on the complete potential energy curves of the ground and first excited states of LiH. Numerical corroboration of the expected continuity for molecular properties for the GVVPT2 method is observed. It is seen that the GVVPT2 electric dipoles are in good agreement with experiment and with other high accuracy theoretical methods.

Configuration-Driven Unitary Group Approach for Generalized Van Vleck Variant Multireference Perturbation Theory

A new, efficient, configuration-driven algorithm utilizing the unitary group approach (UGA) was developed and implemented for the generalized van Vleck perturbation theory (GVVPT) variant of multireference perturbation theory. The computational speed has been improved by 1 or 2 orders of magnitude compared to the previous implementation based on the Table-CI technique. It is shown that the reformulation is applicable to both the second-order (GVVPT2) and third-order (GVVPT3) approximations. Calculations on model problems and on a chemically realistic description of cyclobutadiene are used to illustrate the performance of the method. The calculations on cyclobutadiene, using over 2.3 billion CSFs, provide results on geometric parameters and the barrier height of the automerization reaction in good agreement with established high accuracy results.

Third‐order generalized Van Vleck perturbation theory variant of multireference perturbation theory for electron correlation

AbstractA third‐order approximation to the generalized Van Vleck perturbation theory (GVVPT3) variant of multireference perturbation theory treatment of dynamic electron correlation is described. The approximation builds upon some aspects of the increasingly widely used GVVPT2, which we introduced previously (Khait et al., J Chem Phys 2002, 117, 4133). In addition, using formal relations obtained in connection with GVVPT3, a more thorough understanding of GVVPT2 is obtained, including a key diagnostic that assesses suitability of GVVPT2 for specific tasks. Computational aspects of the implementation of GVVPT3, especially the use of a macroconfiguration description of the model space, are described. The obtained GVVPT3 method can use both complete and incomplete model spaces efficiently. We demonstrate the new method on a series of model problems that demonstrate the applicability of the suggested approach. Among other tests, we show that GVVPT3 is capable of reproducing MRCISD results to within 0.01 eV for the problematic barrier height of the metastable N. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

Generalized van Vleck Perturbation Theory (GVVPT2) Study of the Excited States of Benzene and the Azabenzenes

Generalized van Vleck perturbation theory (GVVPT2) for molecular electronic structures is applied to examine the azabenzene series: benzene, pyridine, pyrazine, symmetric triazine and symmetric tetrazine. The spectra of azabenzenes are complex with large numbers of excited states at low energies comprising n --> pi* and pi --> pi* excited states and also doubly excited states of the n,n --> pi*,pi* type. The calculations are complicated due to strong correlation effects in the nitrogen lone-pair orbitals and the pi electrons. This study is the first to use GVVPT2 on conjugated systems. Comparison is made with experimental data and complete active space second-order perturbation theory, equation of motion coupled cluster and similarity transformed equation of motion coupled cluster theory data. Using polarized valence double split basis sets for benzene and pyrazine (cc-pVDZ) and pyridine (ANO-S) and polarized triple split basis sets (ANO-L) for triazine and tetrazine, the n --> pi* and pi --> pi* states are computed with an average error of 0.28 eV in comparison with available experimental data.

Second-order generalized Van Vleck perturbation theory calculations of potential energy curves for the dissociation of the C–H bond in methane

Ab initio calculations of the potential energy curve for the dissociation of methane have been performed using second-order generalized Van Vleck multireference perturbation theory (GVVPT2), based on CASSCF reference functions, with the cc-pVDZ and cc-pVTZ basis sets. The results show that the GVVPT2 method provides a correct description of the entire curve, including the region of 2.0–3.0 Å, which is known to be problematic even for high-order many-body perturbation theories, including full fourth order. Non-parallelity errors (NPEs) calculated relative to MRCISD show that GVVPT2, MRD-CI extrapolated to full CI, and coupled cluster with iterative triples have curves similar to MRCISD, with NPEs of 4–5 kcal mol−1. The predicted D e of 109.8 kcal mol−1 at the GVVPT2/cc-pVTZ level is in close agreement with that seen at previous high levels of theory, e.g. 109.5 kcal mol−1 at the MRD-CI/6-31G* extrapolated to full CI level and 109.2 kcal mol−1 at the CCSDT-1a/DZP level, and is close to the experimental value of 112.4 kcal mol−1. This study is the first to evidence that the GVVPT2 method would appear to be capable of studying the fragmentation pathways of alkanes.

High-level ab initio calculations on the NiO 2 system

Several high-level ab initio methods were employed in studies of the narrow singlet-triplet separation of the cyclic form of nickel dioxide (NiO 2). It is shown that the complete versions of the locally renormalized coupled cluster method with singles, doubles, and noniterative triples (LR-CCSD(T)) approach, in contrast to the standard CCSD(T) method, provides results in concert with predictions of the density functional theory (DFT) and internally contracted multi-reference configuration interaction method (IC-MRCI), which favor the triplet state to be the lowest one. Relevant discussion of several aspects related to the underlying CCSD calculations, indicate that the dominant role of singly excited amplitudes violates the paradigm about the leading role of two-body effects in the description of the correlation energy. We also show that the multireference perturbation theory, exemplified here by the Generalized Van Vleck Perturbation Theory, requires the use of very large model space in order to properly describe the non-dynamical correlation effects.

Third-order generalized Van Vleck perturbation theory study of the ground and lowest excited states of C 2

Calculations of the potential curves of the ground ( X Σ g + 1 ) and lowest two excited ( B Δ g 1 and B Σ g + ′ 1 ) electronic states of C 2 were performed using a new third-order generalized Van Vleck multireference perturbation theory (GVVPT3), along with the GVVPT2 method and the CASSCF and MRCISD approaches, using the same 6-31G* basis set used in the Abrams and Sherrill full-CI study. Calculations in the range of 1.0–3.0 Å, which spans the equilibrium bond length and the crossing point of the ( X Σ g + 1 ) and B 1 Δ g curves, demonstrate that the deviations of the GVVPT3 curves from the full-CI curves for all states are comparable to MRCISD. GVVPT2 provides significant improvement relative to CASSCF, which already is much improved over even high-order single reference methods. Predictions of the challenging crossing point between the ( X Σ g + 1 ) and B 1 Δ g curves, in the vicinity of the B Σ g + ′ 1 avoided crossing with the ground state, is predicted to milliångstrom accuracy by MRCISD and GVVPT3, while GVVPT2 and CASSCF show deviations of about 0.07 Å.

Low-lying quintet states of the cobalt dimer

Low-lying quintet states of Co2 have been studied using CCSD, CCSD(T), MCSCF, and the Generalized Van Vleck Perturbation Theory (GVVPT2) variant of second-order multireference perturbation theory. Equilibrium bond lengths and harmonic vibrational frequencies of the lowest 5Δ g and states have been calculated and compared with the results of previous studies. A balanced treatment of nondynamic and dynamic correlation effects has been shown to be of crucial importance for the correct description of the considered states. Based on the GVVPT2 calculations, the l5Δ g state is predicted to lie lower than by about 0.3 eV.

Ground and Low-Lying Excited Electronic States of Difluorodiazirine

Ground and low-lying excited electronic states of difluorodiazirine (F2CN2) were investigated using the recently revised, multireference, second-order generalized Van Vleck perturbation theory (GVVPT2). Initial studies on the ground state were carried out with the single-reference Moller−Plesset second order perturbation theory (MP2), quadratic configuration interaction with single and double excitation (QCISD), and coupled cluster with single, double, perturbative triple excitations (CCSD(T)) methods using several correlation-consistent atomic basis sets. MP2 and QCISD did not give semiquantitatively correct descriptions of the ground state of the molecule. For the ground state, GVVPT2 results are in close agreement with CCSD(T) but lie slightly closer to the experimental results. A balanced treatment of nondynamic and dynamic correlation effects has been shown to be of crucial importance for correct description of the system. Equilibrium geometries, excitation energies, and thermodynamic stabilities of ...

Macroconfigurations in molecular electronic structure theory

AbstractBased on the well‐known idea of dividing a molecular orbital set into groups, and imposing restrictions on allowed group occupancies, a more general formalism, which we refer to as the macroconfiguration approach, is developed. The approach is shown to be useful in four complementary contexts: First, as with previous uses of orbital groupings, physical considerations can actively be taken into account in constructing configuration spaces; second, the approach is demonstrated to enable the screening of a vast number of noninteracting electron configuration pairs in a computationally efficient manner; third, all advantages of the Table–configuration interaction (CI) technique can be used effectively for evaluating Hamiltonian matrix elements between interacting configurations; finally, the macroconfiguration technique provides an efficient way of regenerating excited configurations as needed in a calculation. These characteristics indicate that the suggested approach is promising for improving the computational efficiency of methods describing electron dynamic correlation and, in particular, for CI calculations involving highly (e.g., triply and quadruply) excited configurations and for methods of effective Hamiltonian theory. The advantages of the suggested technique are illustrated by the results of calculations on the ground and excited states of the difluorodioxirane (CF2O2) molecule at the second‐order generalized Van Vleck perturbation theory (GVVPT2) level. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

Molecular gradients for the second-order generalized Van Vleck variant of multireference perturbation theory

Recently, a revised second-order generalized Van Vleck perturbation theory (GVVPT2) for the description of molecular electronic structure has been reported [J. Chem. Phys. 117, 4133 (2002)] that is both state selective and of the “perturb-then-diagonalize” type of multireference perturbation theory (MRPT). Herein, formulas for analytic derivatives of the GVVPT2 energy with respect to nuclear perturbations are presented, as are illustrative calculations on model problems. Specifically, it is shown that the modification of the energy denominator, which addresses the so-called intruder-state problem of MRPT, is analytically differentiable with respect to nuclear perturbation and only requires use of matrices available, or directly obtainable, from the underlying multiconfigurational self-consistent field calculation. The developed formalism takes full advantage of the theoretical and computational characteristics of the GVVPT2 energy. In particular, the calculations are performed directly in a spin-adapted basis and utilize the recently introduced concept of macroconfigurations. Moreover, the full flexibility of the energy calculations with respect to arbitrariness of reference—i.e., no restriction to complete active space self-consistent field—is retained. Test calculations on N2 and O3 comparing the analytic derivatives with the results of finite-difference calculations corroborate the formulas and implementation.