Vibrational contributions to cubic response functions from vibrational configuration interaction response theory

In continuation of our recent paper on vibrational quadratic response functions for vibrational configuration interaction wave functions, we present in this paper a derivation and implementation of the pure vibrational cubic response function for vibrational configuration interaction wave functions. In addition, we present combined electronic and vibrational cubic response functions derived from sum-over-states expressions in the Born-Oppenheimer framework and a discussion of complicating issues. The implementation enables analytic calculation of the pure vibrational cubic response function via response theory, which constitutes a part of the vibronic cubic response function.     

Configuration selection as a route towards efficient vibrational configuration interaction calculations

A configuration selective vibrational configuration interaction (CI) approach is presented that efficiently reduces the variational space and thus leads to significant speedups in comparison to standard vibrational CI implementations. Deviations with respect to reference calculations are well below the accuracy of the underlying electronic structure calculations for the potential and hence are essentially negligible. Parallel implementations of the presented configuration selective vibrational CI approaches lead to further significant time savings. Benchmark calculations based on potential energy surfaces of coupled-cluster quality are presented for the fundamental modes of cis- and trans-difluoroethylene. The size-consistency error within the vibrational configuration interaction calculations of the difluoroethylene dimer has been studied in dependence on the excitation level.     

Linear scaling multireference singles and doubles configuration interaction

A linear scaling multireference singles and doubles configuration interaction (MRSDCI) method has been developed. By using localized bases to span the occupied and virtual subspace, local truncation schemes can be applied in tandem with integral screening to reduce the various bottlenecks in a MRSDCI calculation. Among these, the evaluation of electron repulsion integrals and their subsequent transformation, together with the diagonalization of the large CI Hamiltonian matrix, correspond to the most computationally intensive steps in a MRSDCI calculation. We show that linear scaling is possible within each step. The scaling of the method with system size is explored with a system of linear alkane chains and we proceed to demonstrate this method can produce smooth potential energy surfaces via calculating the dissociation of trans-6-dodecene (C(12)H(24)) along the central C[Double Bond]C bond.     

Generalization of analytic energy derivatives for configuration interaction wave functions

This paper takes the form of a review including some original contributions. A fresh derivation of analytic energy derivative expressions for configuration interaction (CI) wave functions is presented. In this method the CI energy is described by _IJC_IC_J(H _IJ-_IJE) so that the orthonormality condition is explicitly included therein. In the sequence of differentiations up to fourth order it will be demonstrated that each derivative may be expressed in terms of (H _IJ-_IJE) and its derivatives in a symmetric way with respect to the interchange of differential variables. In a similar manner, the CI variational condition may be described in an equation which explicitly includes the normalization condition. It is shown that the differentiation of the modified variational condition produces the coupled perturbed configuration interaction (CPCI) equations in directly soluble and compact forms. The necessary formulae for the energy derivatives up to fourth order and the CPCI equations up to second order are explicitly given.     

Dependence of valency on geometry for configuration interaction wave functions

Earlier definitions of valencies of atoms, molecules, and molecular orbitals are extended to configuration interaction (CI ) wave functions. Using these definitions, valencies both at equilibrium and nonequilibrium geometries of molecules are calculated at the CI level and compared with non-CI results. CI valency correlation diagrams are obtained. Valency variation with bond length using correlated wave functions is found to behave properly unlike in the case of SCF wave functions.     

Configuration space partitioning and matrix buildup scaling for the vibrational configuration interaction method

The accurate computation of anharmonic vibrational states for medium to large molecules is a requirement for the detailed understanding of nonlinear multidimensional infrared spectra and the dynamical information encoded in them. The vibrational configuration interaction (VCI) method constitutes a particularly promising tool in this respect. It is generally hampered though by its unfavorable scaling with respect to system size. We analyze the scaling behavior of several well‐known as well as some new approximate VCI schemes in detail, which are complementary to the class of configuration selection schemes developed recently. We find that the combination of a configuration space partitioning, possibly based on configuration selection, with energetic thresholding and resonance screening provides an efficient scheme for the reduction of computational effort involved in VCI calculations while at the same time maintaining sufficient accuracy for the vibrational energies. © 2012 Wiley Periodicals, Inc.     

Electron spin-spin coupling from multireference configuration interaction wave functions

We present the implementation of two-electron spin-spin coupling as a quasidegenerate perturbative treatment of the Breit-Pauli spin-spin Hamiltonian. The evaluation is based on a multireference CI treatment and constitutes one of the first efforts in the calculation of this effect within a highly sophisticated consideration of both nondynamical and dynamical correlation. The extension of existing schemes for efficient calculation, in particular, of the spin-coupling elements necessitated some involved derivations, the outline of which is presented herein. Application of the program to calculations of diagonal as well as off-diagonal spin-coupling elements is illustrated with the test cases O(2) and NH.     

A comparison of two methods for selecting vibrational configuration interaction spaces on a heptatomic system: ethylene oxide

Two recently developed methods for solving the molecular vibrational Schrodinger equation, namely, the parallel vibrational multiple window configuration interaction and the vibrational mean field configuration interaction, are presented and compared on the same potential energy surface of ethylene oxide, c-C(2)H(4)O. It is demonstrated on this heptatomic system with strong resonances that both approaches converge towards the same fundamental frequencies. This confirms their ability to tackle the vibrational problem of large molecules for which full configuration interaction calculations are not tractable.     

A configuration interaction study of phosphine using bonded functions

A computational study is made of the effect of basis set upon the energy, properties and inversion barrier of the phosphine molecule. The calculations are performed at both the SCF and CI level. The flexibility of the double zeta basis is discussed in the light of the results.     

Direct configuration interaction with a reference state composed of many reference configurations

The configuration interaction method where a single reference state is composed of a linear combination of reference configurations is analyzed in detail. In this method single and double replacements are constructed by applying annihilation and creation operators on the reference state. The analysis is based on the recently derived factorization of the direct CI coupling coefficients into internal and external parts. Using the internal coupling coefficients the integrals are transformed to new entities which are used in the diagonalization step. This two-step procedure differs significantly from the usual straightforward one-step direct CI procedure. A number of operations analysis shows that calculations with the present method should be feasible even with a large number of reference configurations in the reference state. Based on first-order perturbation theory the accuracy of the method is predicted to be close to the accuracy obtained with the usual CI method with many reference configurations.     

Self-consistent embedding theory for locally correlated configuration interaction wave functions in condensed matter

We present new developments on a density-based embedding strategy for the electronic structure of localized feature in periodic, metallic systems [see T. Kluner et al., J. Chem. Phys. 116, 42 (2002), and references therein]. The total system is decomposed into an embedded cluster and a background, where the background density is regarded as fixed. Its effect on the embedded cluster is modeled as a one-electron potential derived from density functional theory. We first discuss details on the evaluation of the various contributions to the embedding potential and provide a strategy to incorporate the use of ultrasoft pseudopotentials in a consistent fashion. The embedding potential is obtained self-consistently with respect to both the total and embedded cluster densities in the embedding region, within the framework of a frozen background density. A strategy for accomplishing this self-consistency in a numerically stable manner is presented. Finally, we demonstrate how dynamical correlation effects can be treated within this embedding framework via the multireference singles and doubles configuration interaction method. Two applications of the embedding theory are presented. The first example considers a Cu dimer embedded in the (111) surface of Cu, where we explore the effects of different models for the kinetic energy potential. We find that the embedded Cu density is reasonably well-described using simple models for the kinetic energy. The second, more challenging example involves the adsorption of Co on the (111) surface of Cu, which has been probed experimentally with scanning tunneling microscopy [H. C. Manoharan et al., Nature (London) 403, 512 (2000)]. In contrast to Kohn-Sham density functional theory, our embedding approach predicts the correct spin-compensated ground state.     

Molecular response properties from explicitly time-dependent configuration interaction methods

In this paper we report the calculation of molecular electric response properties with the help of explicitly time-dependent configuration interaction (TD-CI) methods. These methods have the advantage of being applicable (within the limitations of the time-dependent Schrodinger equation) to time-dependent perturbations of arbitrary shape and strength. Three variants are used to solve the time-dependent electronic Schrodinger equation, namely, the TD-CIS (inclusion of single excitations only), TD-CISD (inclusion of single and double excitations), and TD-CIS(D) (single excitations and perturbative treatment of double excitations) methods and applied for illustration to small molecules, H(2) and H(2)O. In the calculation, slowly varying off-resonant electric fields are applied to the molecules and linear (polarizabilities) and nonlinear (hyperpolarizabilities, harmonic generation) response properties are determined from the time-dependent dipole moments.     

Response theory for vibrational wave functions

A formalism for deriving and implementing response functions for vibrational wave functions is described. The formalism utilizes a recently developed second-quantization formulation of many-mode dynamics to define nonredundant parameterizations for different types of approximate vibrational wave functions. The derived response functions cover the cases of an exact state, a vibrational self-consistent field state, and a vibrational configuration interaction state. Sample calculations are presented for the linear-response function and response excitation energies for a two-mode model system and for formaldehyde employing a quartic force field. The advantages and disadvantages of the response theoretical approach for describing excitation energies using different parameterizations are discussed.     

Efficient configuration selection scheme for vibrational second-order perturbation theory

A fast algorithm of vibrational second-order Moller-Plesset perturbation theory is proposed, enabling a substantial reduction in the number of vibrational self-consistent-field (VSCF) configurations that need to be summed in the calculations. Important configurations are identified a priori by assuming that a reference VSCF wave function is approximated well by harmonic oscillator wave functions and that fifth- and higher-order anharmonicities are negligible. The proposed scheme has reduced the number of VSCF configurations by more than 100 times for formaldehyde, ethylene, and furazan with an error in computed frequencies being not more than a few cm(-1).     

The direct configuration interaction method with a contracted configuration expansion

A variant of the configuration interaction method is outlined. There are four characteristic features of the method. First, the freezing, contraction of coefficients in the configuration expansion. In the present approach Rayleigh-Schrödinger perturbation theory has been used for this purpose. Second, the use of the direct CI-method to construct hamiltonian matrix elements between functions with contracted coefficients, in only one sequential read of the molecular integrals. Third, a diagonalization of the resulting small matrix. Fourth, an important computational advantage in that only one CI-vector is needed in core storage instead of two as in the usual direct CI-method. The method has been programmed for the case of all single and double replacements from a spin-unrestricted Hartree-Fock determinant. Test calculations indicate that usually less than 2% of the correlation energy is lost because of the contraction of the CI-expansion. A possibility to avoid the major part of the integral transformation and work directly with atomic basis functions is also discussed.     

Analytic expression of the second derivatives of electronic energy for full configuration interaction wave functions

Summary A new analytic second derivative expression of the electronic energy is derived for full configuration interaction (CI) wave functions. This formula is shown to be free from the derivative terms of both CI and MO coefficients. The second-order relationships between CI and MO coefficients for full CI wave functions are also presented.     

A configuration analysis for fragment interaction

We propose a modified version of configuration analysis (CA) for the fragment interaction in conjunction with Kitaura’s fragment molecular orbital (FMO) scheme. The proposal is abbreviated as CAFI. The MO sets of fragments are merged and then orthonormalized by the use of a weighted Löwdin orthonormalization. The energy calculation is performed with the concurrent electron relaxation functional (CERF). The relaxation energy is obtained in an orbital-wise fashion and is distinguished as the charge-transfer and the polarization. The utility of CAFI is demonstrated through test calculations on hydrogen-bonding systems.     

Efficient methods for configuration interaction calculations

Advanced techniques are developed to provide efficient economic treatment of the large scale eigenvalue problem posed when configuration interaction is carried out on SCF basis sets of moderate size. When the characteristic properties of the hamiltonian matrix are examined in light of the type of solution required, partitioning of the configuration space is shown to result in an expansion of the problem about a limited core of states, where the small but cumulative interactions of vast regions of the remaining space are reduced to the form of an effective potential. With proper selection of the core, the evaluation of this potential can be readily and accurately truncated to a level involving minimum expenditure in time and effort. In particular only diagonal elements and a strip of the full CI matrix are required to achieve an accuracy of 1 – 5 kcal/mole with complete treatment for configuration spaces of order tens of thousands. In addition, a close look at current theory on the generation of matrix elements between spin symmetry adapted configurations leads to simplified expressions where the matrix elements are derived in the form of a weighted sum of molecular integrals in which the weighting coefficients represent the integrated value of the wavefunctions over spin coordinates. For typical cases of low multiplicity and limited numbers of open shells the list of unique parameters needed to generate all weights are shown to be readily stored as a program library. Actual times for matrix element generation are believed to be an order of magnitude faster than current techniques. Practical demonstration of the accuracy and efficiency of the method is provided by calculations on formaldehyde, water, and ethylene.