Analytic techniques for the coupled multiconfiguration SCF perturbation theory

The perturbation theory based on the paired excitation multiconfiguration self-consistent field approach of Clementi and Veillard is considered. The coupled first-order perturbed orbital equations are analysed and an appropriate computational scheme for their solution is discussed. The proposed computational scheme is analogous to the technique employed for the solution of the coupled Hartree–Fock equations in the one-configuration approximation. However, because of the presence ofnondiagonal Lagrangian multipliers and the use of different one-electron operators for different orbitals, the present scheme raises some new computational problems. In this context a new technique for the solution of the unperturbed multiconfiguration self-consistent field equations is proposed. A simple illustration of the superiority of the multiconfiguration perturbation approach with respect to the ordinary coupled Hartree–Fock scheme is given. Also the validity of the variation formulation of the presented scheme and its relation to the finite-field approach are discussed.     

H2-H2 revisited. orthogonalization,polarization and delocalization effects within a bond-orbital scheme

The bond-orbital approach to short-range intermolecular interactions is extended to include polarization and delocalization effects using second-order perturbation theory. Ab initio calculations on H₄ show that results based on a nonorthogonal BO SCF wavefunction closely approach the full MO ones, whereas different orthogonalizatkm procedures give corrections which depend heavily on the accuracy of the one-configuration result.     

Auxiliary density perturbation theory

A new approach, named auxiliary density perturbation theory, for the calculation of second energy derivatives is presented. It is based on auxiliary density functional theory in which the Coulomb and exchange-correlation potentials are expressed by auxiliary function densities. Different to conventional coupled perturbed Kohn-Sham equations the perturbed density matrix is obtained noniteratively by solving an inhomogeneous equation system with the dimension of the auxiliary function set used to expand the auxiliary function density. A prototype implementation for the analytic calculation of molecular polarizabilities is presented. It is shown that the polarizabilities obtained with the newly developed auxiliary density perturbation approach match quantitative with the ones from standard density functional theory if augmented auxiliary function sets are used. The computational advantages of auxiliary density perturbation theory are discussed, too.     

An efficient approach for self-consistent-field energy and energy second derivatives in the atomic-orbital basis

Based on self-consistent-field (SCF) perturbation theory, we recast the SCF and the coupled-perturbed SCF (CPSCF) equations for time-independent molecular properties into the atomic-orbital basis. The density matrix and the perturbed density matrix are obtained iteratively by solving linear equations. Only matrix multiplications and additions are required, and this approach can exploit sparse matrix multiplications and thereby offer the possibility of evaluating second-order properties in computational effort that scales linearly with system size. Convergence properties are similar to conventional molecular-orbital-based CPSCF procedures, in terms of the number of derivative Fock matrices that must be constructed. We also carefully address the issue of the numerical accuracy of the calculated second derivatives of the energy, in order to specify the minimum precision necessary in the CPSCF procedure. It is found that much looser tolerances for the perturbed density matrices are adequate when using an expression for the second derivatives that is correct through second order in the CPSCF error.     

Higher order alchemical derivatives from coupled perturbed self-consistent field theory

We present an analytical approach to treat higher order derivatives of Hartree-Fock (HF) and Kohn-Sham (KS) density functional theory energy in the Born-Oppenheimer approximation with respect to the nuclear charge distribution (so-called alchemical derivatives). Modified coupled perturbed self-consistent field theory is used to calculate molecular systems response to the applied perturbation. Working equations for the second and the third derivatives of HF/KS energy are derived. Similarly, analytical forms of the first and second derivatives of orbital energies are reported. The second derivative of Kohn-Sham energy and up to the third derivative of Hartree-Fock energy with respect to the nuclear charge distribution were calculated. Some issues of practical calculations, in particular the dependence of the basis set and Becke weighting functions on the perturbation, are considered. For selected series of isoelectronic molecules values of available alchemical derivatives were computed and Taylor series expansion was used to predict energies of the "surrounding" molecules. Predicted values of energies are in unexpectedly good agreement with the ones computed using HF/KS methods. Presented method allows one to predict orbital energies with the error less than 1% or even smaller for valence orbitals.     

Quasi-degenerate perturbation theory with general multiconfiguration self-consistent field reference functions

The quasi-degenerate perturbation theory (QDPT) with complete active space (CAS) self-consistent field (SCF) reference functions is extended to the general multiconfiguration (MC) SCF references functions case. A computational scheme that utilizes both diagrammatic and sum-over-states approaches is presented. The second-order effective Hamiltonian is computed for the external intermediate configurations (including virtual or/and core orbitals) by the diagrammatic approach and for internal intermediate configurations (including only active orbitals) by the configuration interaction matrix-based sum-over-states approach. The method is tested on the calculations of excitation energies of H(2)O, potential energy curves of LiF, and valence excitation energies of H(2)CO. The results show that the present method yields very close results to the corresponding CAS-SCF reference QDPT results and the available experimental values. The deviations from CAS-SCF reference QDPT values are less than 0.1 eV on the average for the excitation energies of H(2)O and less than 1 kcal/mol for the potential energy curves of LiF. In the calculation of the valence excited energies of H(2)CO, the maximum deviation from available experimental values is 0.28 eV.     

Magnetic susceptibility of the BH molecule

The coupled Hartree-Fock (CHF) perturbation approach and its extension to multiconfiguration wavefunctions (MC CHF scheme) were used to calculate the magnetic susceptibility of the BH molecule. The results obtained for an SCF and two pair-excitation MC SCF functions confirm the paramagnetism of the BH molecule and indicate a rather weak dependence of the computed molecular susceptibility on the correlation effects. Sponsored by the Institute of Low Temperatures and Structure Research, Polish Academy of Sciences.     

Formulation of magnetically perturbed time-dependent density functional theory

A formulation of time-dependent density functional theory (TDDFT) in the presence of a static imaginary perturbation is derived. A perturbational approach is applied leading to corrections to various orders in the quantities of interest, namely, the excitation energies and transition densities. The perturbed TDDFT equations are relatively straightforward to derive but the resulting expressions are rather cumbersome. Simplifications of these equations are suggested. Both the simplified and full expressions are used to obtain equations for first- and second-order corrections to the excitation energy, the first-order correction to the transition density, and the corrections for both quantities to first-order in two different perturbations. This formulation, called magnetically perturbed TDDFT, details how conventional TDDFT calculations can be corrected to allow for the inclusion of a static magnetic field and/or spin-orbit coupling.     

The MC SCF theory: Method of one-electron Hamiltonian

The one-electron Hamiltonian method is developed to solve the variational equations of the MC SCF theory. The many-parameter family of the one-electron Hamiltonian is derived and conditions for parameters to provide convergence of the SCF procedure to the energy minimum are obtained. A computation scheme based on the use of the one-electron Hamiltonian is described.     

Application of SCF perturbation theory to molecular calculations

A method for solving Roothaan's molecular orbital equations by means of SCF perturbation theory is presented. An estimate of the accuracy of the third order expansion is made for the CNDO/2 approximation from a comparison of the results from direct calculations. It is found that the third order theory is sufficiently accurate for quantitative studies.     

SCF perturbation calculations on metalloporphyrins

The influence of metal ion on the oxidation and ionisation potentials of metalloporphyrins is investigated by the simple electrostatic model using SCF perturbation theory. The zero order wavefunctions are obtained from PPP and CNDO/2 methods. The wide variations in redox potentials with metal and the relative insensitivity of the optical transitions with metal are very well accounted for by the perturbation approach.     

Ab initio surface hopping excited-state molecular dynamics approach on the basis of spin–orbit coupled states: An application to the A-band photodissociation of CH3I

Ab initio molecular dynamics approach has been extended to multi-state dynamics on the basis of the spin–orbit coupled electronic states that are obtained through diagonalization of the spin–orbit coupling matrix with the multi-state second-order multireference perturbation theory energies in diagonal elements and the spin–orbit coupling terms at the state-averaged complete active space self-consistent field level in off-diagonal elements. Nonadiabatic transitions over the spin–orbit coupled states were taken into account explicitly by a surface hopping scheme with utilizing the nonadiabatic coupling terms calculated by numerical differentiation of the spin–orbit coupled wavefunctions and analytical nonadiabatic coupling terms. The present method was applied to the A-band photodissociation of methyl iodide, CH₃I + hv → CH₃ + I (²P_3/2)/I* (²P_1/2), for which a pioneering theoretical work was reported by Amatatsu, Yabushita, and Morokuma. The present results reproduced well the experimental branching ratio and energy distributions in the dissociative products. © 2018 Wiley Periodicals, Inc.     

Molecular orbital theory of the electronic structures of one-dimensional molecular crystals

A non-empirical tight-binding LCAO SCF MO treatment of one-dimensional molecular crystals based on the SCF perturbation theory is presented. The simpler version of this method at the level of the CNDO/2 approximation is also given.     

Thermodynamic properties of model solids with short-ranged potentials from Monte Carlo simulations and perturbation theory

Monte Carlo simulations have been performed to determine the excess energy and the equation of state of fcc solids with Sutherland potentials for wide ranges of temperatures, densities, and effective potential ranges. The same quantities have been determined within a perturbative scheme by means of two procedures: (i) Monte Carlo simulations performed on the reference hard-sphere system and (ii) second-order Barker-Henderson perturbation theory. The aim was twofold: on the one hand, to test the capability of the "exact" MC-perturbation theory of reproducing the direct MC simulations and, on the other hand, the reliability of the Barker-Henderson perturbation theory, as compared with direct MC simulations and MC-perturbation theory, to determine the thermodynamic properties of these solids depending on temperature, density, and potential range. We have found that the simulation data for the excess energy obtained from the two procedures are in close agreement with each other. For the equation of state, the results from the MC-perturbation procedure also agree well with the direct MC simulations except for very low temperatures and extremely short-ranged potentials. Regarding the Barker-Henderson perturbation theory, we have found that in general the second-order approximation does not provide significant improvement over the first-order one.     

A reformulation of the coupled perturbed self-consistent field equations entirely within a local atomic orbital density matrix-based scheme

To exploit the exponential decay found in numerical studies for the density matrix and its derivative with respect to nuclear displacements, we reformulate the coupled perturbed self-consistent field (CPSCF) equations and a quadratically convergent SCF (QCSCF) method for Hartree-Fock and density functional theory within a local density matrix-based scheme. Our D-CPSCF (density matrix-based CPSCF) and D-QCSCF schemes open the way for exploiting sparsity and to achieve asymptotically linear scaling of computational complexity with molecular size (M), in case of D-CPSCF for all (M) derivative densities. Furthermore, these methods are even for small molecules strongly competitive to conventional algorithms.     

SCF perturbation theory for unimolecular reactions

A CNDO/2 SCF perturbation theory is presented for interpreting the form of CNDO/2 potential energy surfaces of unimolecular reactions. The analysis is performed by calculating the energy change E arising from a distortion of the molecular geometry along the reaction coordinate. E is decomposed into different perturbational contributions which are appropriate for an interpretation of the perturbation energy E. Moreover, E is resolved into energy parts arising from a single occupied orbital and contributions due to pairwise orbital interactions. In this way one evaluates numerically how the form of the occupied and unoccupied orbitals determines the magnitude of E. If the distortion occurs along a definite symmetry coordinate, group-theoretical arguments can be applied to discuss the magnitude of characteristic components of the perturbation energy. The SCF perturbation theory is used to analyze the isomerization of ethylene, cis-2-butene and cis-2-butenenitrile.This work was partially supported by Nato-Grant No. 1072     

Hypervirial-perturbative approximation to the SCF eigenvalues of coupled oscillators

The hypervirial relations and perturbation theory are used in order to obtain analytical non-numerical expressions for the SCF eigenvalues of coupled oscillators.     

An efficient molecular orbital approach for a simultaneous evaluation of the vibrational circular dichroism and absorption intensities

We present a procedure for evaluating vibrational circular dichroism and absorption intensities simultaneously, using finite electric field perturbation theory. In this procedure seven SCF calculations are sufficient to obtain all vibrational intensities. Preliminary calculations using CNDO wavefnctions are used to test the soundness of this procedure.     

Analytical evaluation of Fukui functions and real-space linear response function

Many useful concepts developed within density functional theory provide much insight for the understanding and prediction of chemical reactivity, one of the main aims in the field of conceptual density functional theory. While approximate evaluations of such concepts exist, the analytical and efficient evaluation is, however, challenging, because such concepts are usually expressed in terms of functional derivatives with respect to the electron density, or partial derivatives with respect to the number of electrons, complicating the connection to the computational variables of the Kohn-Sham one-electron orbitals. Only recently, the analytical expressions for the chemical potential, one of the key concepts, have been derived by Cohen, Mori-Sánchez, and Yang, based on the potential functional theory formalism. In the present work, we obtain the analytical expressions for the real-space linear response function using the coupled perturbed Kohn-Sham and generalized Kohn-Sham equations, and the Fukui functions using the previous analytical expressions for chemical potentials of Cohen, Mori-Sánchez, and Yang. The analytical expressions are exact within the given exchange-correlation functional. They are applicable to all commonly used approximate functionals, such as local density approximation (LDA), generalized gradient approximation (GGA), and hybrid functionals. The analytical expressions obtained here for Fukui function and linear response functions, along with that for the chemical potential by Cohen, Mori-Sánchez, and Yang, provide the rigorous and efficient evaluation of the key quantities in conceptual density functional theory within the computational framework of the Kohn-Sham and generalized Kohn-Sham approaches. Furthermore, the obtained analytical expressions for Fukui functions, in conjunction with the linearity condition of the ground state energy as a function of the fractional charges, also lead to new local conditions on the exact functionals, expressed in terms of the second-order functional derivatives. We implemented the expressions and demonstrate the efficacy with some atomic and molecular calculations, highlighting the importance of relaxation effects.