BONDED TABLEAU AND UNITARY GROUP APPROACH FOR MANY-ELECTRON PROBLEMS

A unitary group approach based on the so-called bonded tableaux (VB) states is described. Several different realizations of the bonded tableaux are discussed and their relations are pointed out. From a viewpoint of the symmetric group we reveal a simple structure of the matrix elements of unitary group generators and generator products. This structure makes an efficient approach to the matrix element evaluation.     

化学子体系相互作用理论的键表方法

本文应用作者最近提出的多电子体系键表方法研究化学子体系的相互作用理论. 提出了两种构造子体系对称性匹配分割波函数的方法, 即置换群非标准基投影算子的方法和键表耦合的方法. 在此基础上研究子体系相互作用能的计算. 应用本文的方法可在价键(VB)结构相互作用或组态相互作用(CI)水平上处理各种化学上重要的多电子体系(原子或分子)的分割及其相互作用问题, 如分子间的或分子内的基团或壳层的相互作用。     

Construction and applications of symmetrized valence bond wave functions

A method constructing symmetry-adapted bonded Young tableau bases is proposed, based on the symmetry properties of bonded tableaus and the projection operator associated with a point group. Several examples including the ground states and π excited states of O₃⁻, O₃, O₃⁺, and C₃⁻ are shown for instruction to construct the symmetrized valence bond (VB) wave function. Excitation energies of transitions from the ground states to π excited states of O₃⁻, C₃H₅, and C₃⁻ are calculated with an optimized symmetrized valence bond wave function in the σ–π separation approximation. Good agreement between the VB and experimental excitation energies is observed. The bonding features of the ground state and the first π excited singlet and triplet states for S₃ are discussed according to bonding populations from VB calculations. Both the singlet-biradical and the dipole structures have significant contributions to the ground state X ¹A₁ of S₃, while the excited state 1 ¹B₂ is essentially composed of the dipole structures, and the 1 ³B₂ excited state is comprised from a triplet-biradical structure. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 66 : 1–7, 1998     

价键理论新进展

概要介绍了现代价键理论的几个主要方法,并讨论了它们各自的特点及其发展现状,并重点介绍了键表方法的基本理论、计算程序及一些应用。     

Valence bond approach exploiting Clifford algebra realization of Rumer–Weyl basis

A detailed algorithm is described that enables an implementation of a general valence bond (VB ) method using the Clifford algebra unitary group approach (CAUGA ). In particular, a convenient scheme for the generation and labeling of classical Rumer–Weyl basis (up to a phase) is formulated, and simple rules are given for the evaluation of matrix elements of unitary group generators, and thus of any spin-independent operator, in this basis. The case of both orthogonal and nonrothogonal atomic orbital bases is considered, so that the proposed algorithm can also be exploited in molecular orbital configuration interaction calculations, if desired, enabling a greater flexibility for N-electron basis-set truncation than is possible with the standard Gel'fand–Tsetlin basis. Finally, an exploitation of this formalism for the VB method, based on semiempirical Pariser–Parr–Pople (PPP )-type Hamiltonian and nonorthogonal overlap-enhanced atomic orbital basis, and its computer implementation, enabling us to carry out arbitrarily truncated or full VB calculations, is described in detail.     

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.     

Review of biorthogonal coupled cluster representations for electronic excitation

Single-reference coupled-cluster (CC) methods for electronic excitation are based on a biorthogonal representation (bCC) of the (shifted) Hamiltonian in terms of excited CC states, also referred to as correlated excited (CE) states, and an associated set of states biorthogonal to the CE states, the latter being essentially configuration interaction (CI) configurations. The bCC representation generates a non-hermitian secular matrix, the eigenvalues representing excitation energies, while the corresponding spectral intensities are to be derived from both the left and right eigenvectors. Using the perspective of the bCC representation, a systematic and comprehensive analysis of the excited-state CC methods is given, extending and generalizing previous such studies. Here, the essential topics are the truncation error characteristics and the separability properties, the latter being crucial for designing size-consistent approximation schemes. Based on the general order relations for the bCC secular matrix and the (left and right) eigenvector matrices, formulas for the perturbation-theoretical order of the truncation errors (TEO) are derived for energies, transition moments, and property matrix elements of arbitrary excitation classes and truncation levels. In the analysis of the separability properties of the transition moments, the decisive role of the so-called dual ground state is revealed. Due to the use of CE states, the bCC approach can be compared to so-called intermediate state representation (ISR) methods based exclusively on suitably orthonormalized CE states. As the present analysis shows, the bCC approach has decisive advantages over the conventional CI treatment, but also distinctly weaker TEO and separability properties in comparison to a full (and hermitian) ISR method.     

H_3~+体系基态和电子激发态的价键方法研究

本文应用价键理论的键表酉群方法,讨论H_3~+体系的基态及其几个低激发态,研究了该体系处于激发态时价键结构的变化,得到H_3~+体系的基态为等边三角形,第一激发态为直线型,第二、三激发态为等腰三角形,且基本简并.     

Unitary group approach to the many-electron problem. II. Adjoint tensor operators for U(n)

In this paper we present a derivation of the U(n) adjoint coupling coefficients for the representations appropriate to many-electron systems. Since the states of a many-fermion system are to comprise the totally antisymmetric Nth rank tensor representation of U(2n), the work of this paper enables the matrix elements of the U(2n) generators to be evaluated directly in the U(n) × U(2) (i.e., spin orbit) basis using their transformation properties as adjoint tensor operators. A connection between the adjoint coupling coefficients, as derived in this paper, and the matrix elements of certain (spin independent) two-body operators is also presented. This indicates that in CI calculations, one may obtain the matrix elements of spin-dependent operators from the known matrix elements of certain spin-independent two-body operators. In particular this implies a segment-level formula for the matrix elements of the U(2n) generators in the spin-orbit basis.     

Efficient method for computation of representation matrices of the unitary group generators

The representation matrices of the unitary group generators E_k1 are equivalent to the representation matrices of cyclic permutations (k, k ± l, …, l ? 1, l). A method is presented for simultaneous computation of matrices corresponding to all different generators at a cost of less than one multiplication per nonzero element. The number of operations necessary for calculation of individual matrices for single generators or for products of two generators is at most proportional to the number of matrix elements of the final matrix. This approach eliminates the need to store the representation matrices in CI calculations.     

价键结构函数与键数Ⅰ.理论处理及甲烷的应用实例

采用键函数形式对键表进行逐级展开,其中荚函数的强正交积作为零级函数,进而构造了各级校正函数以描述多对键之间的相关作用,同时采用Heitler-london 函数作为键函数,对甲烷进行离键表计算,结果表明CH~4中一.二.三级校正能近似有4:2:-1的关系,即校正函数的收敛性能较好.     

Valence bond theoretical study for chemical reactivity

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Unitary group approach to the many-electron problem. I. Matrix element evaluation and shift operators

This is the first paper in a series of three directed toward the evaluation of spin-dependent Hamiltonians directly in the spin-orbit basis. In this paper we present a new and complete derivation of the matrix elements of the U(n) generators in the electronic Gel'fand basis. The approach employed differs from previous treatments in that the matrix elements of nonelementary generators are obtained directly. A general matrix element formula is derived which explicitly demonstrates the segment level formalism obtained previously by Shavitt using different methods. A simple relationship between the matrix elements of raising and lowering generators is determined which indicates that in CI calculations, only the matrix elements of raising generators need be calculated. Some results on the matrix elements of products of two generators are also presented.     

H3势能面的键表方法计算

本文采用键表自洽场方法计算H₃体系的势能面,完成了用键表相互作用和多键表自洽场方法计算H₃体系的程序编写和计算,讨论了各种近似条件下键表方法的特点,得到一些对价键方法应用具有建设性意义的结论。     

A diabatic representation including both valence nonadiabatic interactions and spin-orbit effects for reaction dynamics

A diabatic representation is convenient in the study of electronically nonadiabatic chemical reactions because the diabatic energies and couplings are smooth functions of the nuclear coordinates and the couplings are scalar quantities. A method called the fourfold way was devised in our group to generate diabatic representations for spin-free electronic states. One drawback of diabatic states computed from the spin-free Hamiltonian, called a valence diabatic representation, for systems in which spin-orbit coupling cannot be ignored is that the couplings between the states are not zero in asymptotic regions, leading to difficulties in the calculation of reaction probabilities and other properties by semiclassical dynamics methods. Here we report an extension of the fourfold way to construct diabatic representations suitable for spin-coupled systems. In this article we formulate the method for the case of even-electron systems that yield pairs of fragments with doublet spin multiplicity. For this type of system, we introduce the further simplification of calculating the triplet diabatic energies in terms of the singlet diabatic energies via Slater's rules and assuming constant ratios of Coulomb to exchange integrals. Furthermore, the valence diabatic couplings in the triplet manifold are taken equal to the singlet ones. An important feature of the method is the introduction of scaling functions, as they allow one to deal with multibond reactions without having to include high-energy diabatic states. The global transformation matrix to the new diabatic representation, called the spin-valence diabatic representation, is constructed as the product of channel-specific transformation matrices, each one taken as the product of an asymptotic transformation matrix and a scaling function that depends on ratios of the spin-orbit splitting and the valence splittings. Thus the underlying basis functions are recoupled into suitable diabatic basis functions in a manner that provides a multibond generalization of the switch between Hund's cases in diatomic spectroscopy. The spin-orbit matrix elements in this representation are taken equal to their atomic values times a scaling function that depends on the internuclear distances. The spin-valence diabatic potential energy matrix is suitable for semiclassical dynamics simulations. Diagonalization of this matrix produces the spin-coupled adiabatic energies. For the sake of illustration, diabatic potential energy matrices are constructed along bond-fission coordinates for the HBr and the BrCH(2)Cl molecules. Comparison of the spin-coupled adiabatic energies obtained from the spin-valence diabatics with those obtained by ab initio calculations with geometry-dependent spin-orbit matrix elements shows that the new method is sufficiently accurate for practical purposes. The method formulated here should be most useful for systems with a large number of atoms, especially heavy atoms, and/or a large number of spin-coupled electronic states.     

Preparation and resolution of molecular states by coherent sequences of phase-locked ultrashort laser pulses

We study the application of nonlinear wave packet interferometry to the preparation and resolution of the overlaps of nonstationary nuclear wave functions evolving in an excited electronic state of a diatomic molecule. It is shown that possible experiments with two phase-locked ultrashort pulsepairs can be used to determine a specific vibrational wave packet state in terms of coherent states of the ground electronic state. We apply this scheme to an idealized molecule with harmonic potential energy surfaces and to the X <-- B transition states of the iodine molecule. Our results indicate that this scheme is very promising as a potential tool to quantum control.     

Unitary group approach to spin-adapted open-shell coupled cluster theory

We show that the irreducible tensor operators of the unitary group provide a natural operator basis for the exponential Ansatz which preserves the spin symmetry of the reference state, requires a minimal number of independent cluster amplitudes for each substitution order, and guarantees the invariance of the correlation energy under unitary transformations of core, open-shell, and virtual orbitals. When acting on the closed-shell reference state with n_c doubly occupied and n_v unoccupied (virtual) orbitals, the irreducible tensor operators of the group U(n_c) ? U(n_V) generate all Gelfand-Tsetlin (GT) states corresponding to appropriate irreducible representation of U(n_c + n_v). The tensor operators generating the M-tuply excited states are easily constructed by symmetrizing products of M unitary group generators with the Wigner operators of the symmetric group S_M. This provides an alternative to the Nagel-Moshinsky construction of the GT basis. Since the corresponding cluster amplitudes, which are also U(n_c) ? U(n_s) tensors, can be shown to be connected, the irreducible tensor operators of U(n_c) ? U(n_v) represent a convenient basis for a spin-adapted full coupled cluster calculation for closed-shell systems. For a high-spin reference determinant with n, singly occupied open-shell orbitals, the corresponding representation of U(n), n=n_c + n_v + n_s is not simply reducible under the group U(n_c) ? U(n_s) ? U(n_v). The multiplicity problem is resolved using the group chain U(n) ? U(n_c + n_v) ? U(n_s) ? U(n_c) ?U(n_s)? U(n_v) ? U(n_v). The labeling of the resulting configuration-state functions (which, in general, are not GT states when n_c > 1) by the irreducible representations of the intermediate group U(n_c + n_v) ?U(n_s) turns out to be equivalent to the classification based on the order of interaction with the reference state. The irreducible tensor operators defined by the above chain and corresponding to single, double, and triple substitutions from the first-, second-, and third-order interacting spaces are explicitly constructed from the U(n) generators. The connectedness of the corresponding cluster amplitudes and, consequently, the size extensivity of the resulting spin-adapted open-shell coupled cluster theory are proved using group theoretical arguments. The perturbation expansion of the resulting coupled cluster equations leads to an explicitly connected form of the spin-restricted open-shell many-body perturbation theory. Approximation schemes leading to manageable computational procedures are proposed and their relation to perturbation theory is discussed. © 1995 John Wiley & Sons, Inc.     

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.