Spin-Dependent operators in the spin-free quantum chemistry

The two-particle spatial density matrix components introduced by McWeeny are expressed in terms of the Fock coordinate wave function, which is constructed from an arbitrary function of N spatial coordinates. The integral relations for these components are verified. The necessary matrix elements of a standard representation of the S_N group are calculated.     

The non-N-representability of the Colle–Salvetti second-order reduced density matrix

The Colle–Salvetti second-order reduced density matrix (2-matrix) is an approximation to the 2-matrix obtained from a wave function that is a product of a reference wave function containing little or no correlation times a product of correlation factors that are functions of the coordinates of pairs of electrons. A formal proof is given for the non-N-representability for the Colle–Salvetti 2-matrix using the nonnegativity condition of the 2-matrix. The nonnegativity condition of the particle-hole overlap matrix (G matrix) is also not satisfied. The proof is valid for Colle–Salvetti 2-matrices obtained from both the Hartree–Fock and small multiconfigurational-self-consistent-field wave functions. Even though the Colle–Salvetti 2-matrix is not N-representable, it does satisfy the Pauli principle component of the G-matrix condition because it reduces to an N-representable first-order reduced density matrix. © 1993 John Wiley & Sons, Inc.     

Discrete variable representation method applied to the determination of rotation-vibration bound states of NO2

The discrete variable representation method is applied to the determination of the rotation-vibration energy levels of the fundamental electronic state of NO₂. The Hamiltonian is expressed in Johnson hyperspherical coordinates and developed on a DVR basis for each internal coordinate, while parity-adapted linear combinations of Wigner functions are used to describe the rotational motion. The diagonalization of the Hamiltonian matrix is performed using the Lanczos algorithm for large symmetric and Hermitian matrices. Results for rovibrational states up to J = 11 for the first five vibrational energy levels are presented. © 1997 John Wiley & Sons, Inc.     

Ab initio calculations of the ultraviolet resonance Raman spectra of uracil

An equation been derived to calculate, ab initio, the frequencies and intensities of a resonant Raman spectrum from the transform theory of resonance Raman scattering. This equation has been used to calculate the intensities of the ultraviolet resonance Raman spectra from the first π-π* excited state of uracil and 1,3-dideuterouracil. The protocol for this calculation is as follows: (1) The force constant matrix elements in Cartesian coordinate space, the vibrational frequencies, and the minimum energy ground and excited state geometries of the molecule are calculated ab initio using the molecular orbital program Gaussian 92, (2) the force constants in Cartesian coordinates are transformed into force constants in the space of a set of 3N – 6 nonredundant symmetrized internal coordinates, (3) the G matrix is constructed from the energy minimized ground state Cartesian coordinates and the GFL = LΛ eigenvalue equation is solved in internal coordinate space, (4) the elements of the L and L^?1 matrices are calculated, (5) the changes in all of the internal coordinates in going from the ground to the excited state are calculated, and (6) these results are used in combination with the transform theory of resonance Raman scattering to calculate the relative intensities of each of the 3N – 6 vibrations as a function of the exciting laser frequency. There are no adjustable parameters in this calculation, which reproduces the experimental frequencies and intensities with remarkable fidelity. This indicates that the Dushinsky rotation of the modes in the excited state of these molecules is not important and that the simplest form of the transform theory is adequate. © 1995 John Wiley & Sons, Inc.     

Efficient evaluation of the algebrants of VB wave functions using the successive expansion method. I. Spin S = 0, ½

An efficient expansion method for the evaluation of VB matrix elements is introduced. The overlaps of VB wave functions of N electrons can be treated as algebrants, i.e., generalized determinants, of N × N matrices. An algebrant can be expanded with subalgebrants of lower orders in a successive way. By choosing Rumer spin bases and appropriately arranging the expansion, it is found that the number of unique subalgebrants involved in the expansion increases in a quite moderate way with N. In contrast to the traditional symmetric group approach, which explicitly utilizes all N! representation matrices, the new strategy incorporates the group theoretical factors in a simple way in the successive expansion. As only the unique subalgebrants are further expanded, the computational effort required by the new strategy scales in a very acceptable manner with the increasing number of electrons. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 62: 245–259, 1997     

He++N2 Charge Transfer Reaction: An Ab initio Analysis

An ab initio analysis on the involved potential energy surfaces is presented for the investigation of the charge transfer mechanism for the He⁺+N₂ system. At high collision energy, as many as seven low-lying electronic states are observed to be involved in the charge transfer mechanism. Potential energy surfaces for these low-lying electronic states have been computed in the Jacobi scattering coordinates, applying multireference configuration interaction level of theory and aug-cc-pVQZ basis sets. Asymptotes for the ground and various excited states are assigned to mark the entrance (He⁺+N₂) and charge transfer channels (He+N₂⁺). Nonadiabatic coupling matrix elements and quasi-diabatic potential energy surfaces have been computed for all seven states to rationalize the available experimental data on the charge transfer processes and to facilitate dynamics studies.     

The Gaussian and augmented-plane-wave density functional method for ab initio molecular dynamics simulations

A new algorithm for density-functional-theory-based ab initio molecular dynamics simulations is presented. The Kohn–Sham orbitals are expanded in Gaussian-type functions and an augmented-plane-wave-type approach is used to represent the electronic density. This extends previous work of ours where the density was expanded only in plane waves. We describe the total density in a smooth extended part which we represent in plane waves as in our previous work and parts localised close to the nuclei which are expanded in Gaussians. Using this representation of the charge we show how the localised and extended part can be treated separately, achieving a computational cost for the calculation of the Kohn–Sham matrix that scales with the system size N as O(NlogN). Furthermore, we are able to reduce drastically the size of the plane-wave basis. In addition, we introduce a multiple-cutoff method that improves considerably the performance of this approach. Finally, we demonstrate with a series of numerical examples the accuracy and efficiency of the new algorithm, both for electronic structure calculations and for ab initio molecular dynamics simulations. Received: 15 December 1998 /Accepted: 18 February 1999 /Published online: 14 July 1999     

Note on the space part of anti-symmetric wave functions in the many-electron problem

It is pointed out that if a many-electron antisymmetric wave function is expanded as a sum of spin-product functions, each multiplied by a function of coordinates, the resulting functions of coordinates have many of the same useful features found with the symmetric and antisymmetric functions representing singlet and triplet states in a two-electron system. For finding the energy, or any function of coordinates only, in the approximation in which spin-orbit interaction is neglected, one such function of coordinates can be used, the spins being disregarded. Simple procedures allow one to find matrix components of such operators as S ² and L . S from the functions of coordinates. These procedures are much easier to visualize than the use of projection operators, the permutation group, or other methods in current use. The general procedures are illustrated by application to the three-electron problem of the lithium atom, as treated by Lunell, Kaldor, and Harris, and their application to the contact hyperfine structure is pointed out.     

Generic shapes for the conformation analysis of macrocyclic structures

A new algorithm for the systematic generation of conformations of macrocyclic systems is presented. The procedure is based on the concept of generic shapes that are found in such structures. These shapes are characterized by a selection of harmonics which occur in an approximate Fourier representation of the atomic coordinates of the rings. Following a fixed protocol, a limited set of in-plane and out-of-plane circular harmonics is used to define an ensemble of generic ring shapes. These generic shapes are used as start structures for energy minimizations by a given force-field method. To account for the possibility of having several final conformations originating from the same generic shape, the corresponding initial structure is taken several times and subjected to a randomization step before minimization. The resulting conformations that fall within a preset low-energy band are collected and screened for duplicates and enantiomers. The efficiency of this procedure (ratio between the number of accepted conformations and the total number of energy minimizations) depends on the flexibility of the macrocyclic system. The efficiency is generally quite high for very flexible rings. According to the proposed protocol, the number of generic shapes used as start structures grows as the square of N(lnN), where N is the ring size. The algorithm lends itself to conformational analyses of medium-size and large rings as well as of loops spanned between fixed structural units.     

A method for the calculation of normal-mode vibrational frequencies using symmetry coordinates. Application to the calculation of secondary deuterium isotope effects on carbocations

A method is introduced for the calculation of normal-mode vibrational frequencies of polyatomic molecules based on numerical differencing of analytical gradients in symmetry coordinates. This procedure requires a number of gradient evaluations equal to the largest number of symmetry coordinates belonging to any single irreducible representation of the molecular point group (plus a single gradient evaluation at the equilibrium configuration), which is fewer than the 3N-6 (N atoms) gradient evaluations needed for schemes based on Cartesian or internal coordinates. While the proposed method will not generally be as efficient as procedures which involve the direct calculation of energy second derivatives analytically (as are now available for single-determinant wavefunctions) it appears to be equally accurate, and it should be the method of choice for frequency calculations involving multideterminant wavefunctions for which analytical second-derivative algorithms have yet to be developed. The method is illustrated by the calculation of equilibrium secondary deuterium-isotope effects on a number of reactions involving simple carbocations.     

Exact calculations of Franck–Condon overlaps and of matrix elements of some potentials by means of the coherent state representation

Franck–Condon overlaps are described as the matrix elements of unitary operators related to the spatial displacement and the frequency shift. They are calculated exactly by means of the coherent state representation. Furthermore, the generalized matrix elements of x^j, e^?αx, and e^?βx₂ between two states with different equilibrium coordinates and frequency are evaluated in the same way.     

Spin-free approach for evaluation of electronic matrix elements using character operators of ℒN

A spin-free method is presented for evaluating electronic matrix elements over a spin-independent many-electron Hamiltonian. The spin-adapted basis of configuration state functions is obtained using a nonorthogonal spin basis consisting of projected spin eigenfunctions. The general expressions for the matrix elements are given explicitly, and it is demonstrated how the matrix elements may be obtained simply from the knowledge of the irreducible characters of the permutation group ℒ_N. The presented formulas are very general and may be applied in connection with both spin-coupled valence bond studies and in conventional configuration interaction (CI) methods based on an orthonormal orbital basis. © 1996 John Wiley & Sons, Inc.     

Fluctuation free multivariate integration based logarithmic HDMR in multivariate function representation

This paper focuses on the Logarithmic High Dimensional Model Representation (Logarithmic HDMR) method which is a divide–and–conquer algorithm developed for multivariate function representation in terms of less-variate functions to reduce both the mathematical and the computational complexities. The main purpose of this work is to bypass the evaluation of N–tuple integrations appearing in Logarithmic HDMR by using the features of a new theorem named as Fluctuationlessness Approximation Theorem. This theorem can be used to evaluate the complicated integral structures of any scientific problem whose values can not be easily obtained analytically and it brings an approximation to the values of these integrals with the help of the matrix representation of functions. The Fluctuation Free Multivariate Integration Based Logarithmic HDMR method gives us the ability of reducing the complexity of the scientific problems of chemistry, physics, mathematics and engineering. A number of numerical implementations are also given at the end of the paper to show the performance of this new method.     

Spectrum of states in icosahedral structures of the electronic configuration gn (N = 1–7). 1. Invariant expansion for integrals and energies

The energy spectrum of the states that appear in structures of icosahedral (I,I_h symmetry with open electronic shells g^N (dim g = 4; N = 1–7) is reported. The energies are obtained in terms of integral invariants (reduced matrix elements of electron-electron interaction) H^k (g, g). The latter are analogs of the Slater-Condon parameters F^k(l,l) for atoms with the l^N electronic configuration. A similar representation is proposed for the integrals mm’≨’) of electron-electron interaction on the 4-fold degenerate g orbitals in the “standard” representation. The relation between the terms of the g^N(I,I_h) configuration and the parent states of the orthogonal group O⁺(4) is discussed. Translated fromZhurnal Strukturnoi Khimii, Vol. 38, No. 1, pp. 3–13, January–February, 1997.     

Valence-Bond calculations and matrix elements between two tableau functions of non-orthogonal orbitals

An integral formula is derived involving a new type of determinantal function which reproduces the effect of the Young operator θNPN. This result is used to calculate matrix elements between tableau functions of non-orthogonal orbitals. Matrices which transform this representation into the traditional valence-bond scheme are also given.     

Intramolecular interaction effects and the structure of N2OS and N2O2: An Ab initio study

An ab initio study of O?N? N?S with full geometry optimization has been carried out to corroborate the presence of an interaction between the terminal atoms in this type of structure, which, in O?N? N?O, apparently stabilizes the cis conformer. Using the unscaled 4–31G basis set with a full set of d functions on the sulfur, there is a potential minimun at the trans but not the cis geometry. A gauche conformer with a torsional angle of 77.2° is the most stable. With N₂O₂ this basis set gives potential minima at both the cis and trans geometries, but the trans conformer is slightly more stable, contrary to experiment and the results of (7,3) basis-set calculations reported in the literature in which Gaussian lobe functions were employed. Using a (9,5) basis set there is no longer a potential minimum at the cis geometry, and a gauche structure is more stable than the cis conformer as in the case of N₂OS with the less-extended basis set. Force constants (harmonic and anharmonic), compliance constants, relaxed force constants, and interaction-displacement coordinates for both molecules are compared for key structural elements.