群的对称性偶合系数计算的新方法

<正> 应用广义Wigner—Eckart定理于简化分子体系的量子化学计算可把低对称群微扰理论中矩阵元的计算还原为高对称群矩阵元的计算。这里涉及到高对称群一低对称群和低对称群V系数的计算。人们通常采用投影算符法计算V系数,然后计算高对称性的偶合系数。许多工作者曾就此作了有益的讨论和详细的计算。     

对称性轨道的构造

提出1种以配体轨道的旋转性质为基础,并利用群生成元表示矩阵来构造对称性轨道的新方法。讨论了对称性系数之间的相互关系,同时给出群重迭积分的计算过程。     

对称轨道及张量的新理论方法

提出了产生对称轨道 (symmetricorbital,SO)的标准方法与封闭公式 .SO对于乘法运算是封闭的 ,SO的直积构成N秩SO张量 (NthrankSOtensor,SOT)群 .在进行高秩张量矩阵元计算时 ,SOT方法能自动实现物理因子与几何因子的完全分离 ,无需偶合系数 .与不可约张量方法比较 ,计算过程大大简化 .     

建立在群对称轨道(SMO)及特征组态(CD)基础上的CI方法与程序发展

ISMO－CDCI方法点群理论，特别是不可约张量方法，在量子化学理论方法发展以及简化概念与计算方面，发挥了重要作用．但在国外的量子化学计算程序（如G94）中，在后自洽场（opt－SCF）计算方面，很少用到对称约化．在文献中，只见有关hbeltah群（DZh与它的子群）对称约化用于组态相关计算的报导．由于多重耦合系数的计算复杂，蜕化不可约表示多体问题的对称约化难于得到解决．我们提出了一个统一与普遍的方法，它能解决所有分子体系多体相关的点群对称约化问题［‘－6］，这个方法的核心是群对称轨道（SMO）概念的提出．SMO的基本特…     

多电子体系键表的酉群方法

本文提出了一种对称性匹配无自旋价键型函数,用键表来表示,对多电子体系采用键表为基进行酉群方法处理。正则键表构成完备集,其Hamilton矩阵元的计算可由酉群生成元及乘积对键表的作用而约化为键表的重叠积分的计算来实现。本文提出的键表酉群方法(BTUGA)对计算酉群生成元及多重乘积的矩阵元是十分简便的。     

d^N离子在五角场中的基矢和有关的矩阵元

本文证明了,若选定与群键(SO_3D_nC_n)×SU(2)对称性相匹配的单电子基矢,则由它们直积构成的单个Slater函数是d~N体系D_nC_n旋量群不可约表示的基矢,或其2个一维表示直和的基矢。讨论了D_5对称体系对称基矢及有关矩阵元和顺磁参数计算公式。     

C_(3v)晶场中d~3离子吸收谱的理论研究

为了计算Cr~(3+)离子在C_(3v)对称晶场中的能谱,本文采用一种形式介于Richardson和Watson SCF波函数之间的束缚态d电子径向轨函;从Oh点群的Griffith标准基出发,用一种简便的方法,推出d~3系的C_(3v)点群强场谱项能量矩阵。由此计算了Ruby(Al_2O_3+Cr~(3+))和Cr_2O_3晶体中Cr~(3+)离子的吸收谱。在考虑扳——电耦合后,理论结果与实验符合甚好,从而解释了这两类晶体中Cr~(2+)吸收光谱的机制。这种计算方法既使拟合参数的数目大为减少,又使配位场理论计算格地理论化。     

I群-D2群V系数

本文讨论了I群-D₂群V系数,给出在I群单值表示下的计算值。将以I群为出发点的群分解链延伸到其D₂子群。还列出了D₂群V系数的计算结果。     

分子振动力常数计算中的Λ矩阵和F矩阵的性质

在由G矩阵因解和频率确定力常数的计算中,找到了适用于任意维A矩阵相对于L矩阵按对称坐标排列顺序的关系;用逆酉变换得到了由复合力常数F_s计算F_R的一般方法;确立了F_R按对称块的加和性。以尿素等分子为例,用一般价力场计算了全部力常数F_R,频率计算值(低频部分)更符合实验结果。     

d~5系C_(2h)点群强场方案和MnCl_2·2H_2O的吸收谱

利用一种处理低对称络离子吸收谱强场方案,推出d~5系C_(2h)点群不可约表示对应的能量矩阵。考虑振动电子偶合及双电子激发,解释了MnCl_2·2H_2O晶体中Mn~(2+)的复杂吸收谱。确认了Mn~(2+)络离子晶场的C_(2h)对称性。     

The symmetrized random matrix approach, an efficient method to obtain relativistic molecular symmetry adapted functions

In relativistic quantum chemical calculation of molecules, where the spin-orbit interaction is included, the electron orbitals possess both the double point group symmetry and the time-reversal symmetry. If symmetry adapted functions are employed as the basis functions of electron orbitals, it would allow a significant reduction of the computational expense. The point group symmetry adapted functions can be obtained by the group projection operators via its actions on the atomic orbital functions. We have proposed an efficient and simple method to obtain all irreducible representation matrices, which are the basis of the group projection operators, of any finite double point group. Both double point group symmetry and time-reversal symmetry are automatically imposed on the representation matrices. This is achieved by the symmetrized random matrix (SRM) approach, where the SRM is constructed in the regular representation space of a finite group and the eigenfunctions of SRM provide all irreducible representation matrices of the given point group.     

The symmetric groups and the Waller–Hartree spin-free method

The relations between the Waller–Hartree spin-free method and the symmetric group theory are given. It is shown that the Gallup method is a special case of ours with S = M. Furthermore, all the irreducible representation matrices and other matrices needed are written explicitly in terms of Sanibel coefficients which makes the method more useful. However, it was shown that the cases with S ≠ M for the spin-free pure spin states might be beyond the power of the symmetric group theory.     

Elements of irreducible tensorial matrices generated by finite groups with applications to ligand field Hamiltonians

Using symmetry to determine Hamiltonian matrix elements for quantum systems with finite group symmetry is a special case of obtaining group-generated irreducible tensorial matrices. A group-generated irreducible tensorial matrix transforms irreducibly under the group and is a linear combination of group transformations on a reference matrix. The reference matrix elements may be appropriate integrals or parameters. The methods of normalized irreducible tensorial matrices (NITM) are employed to express elements of the generated matrix in terms of those of the reference matrix without performing the actual transformations. Only NTTM components of the reference matrix with the same transformation properties as the group-generated matrix will contribute to its elements. The elements of invariant symmetry-generated matrices are proportional to simple averages of certain elements of the reference matrix. This relation is substantially more efficient than previous techniques for evaluating matrix elements of octahedral and tetragonal d-type ligand-field Hamiltonians.     

Some simplification in spin-free configuration interaction studies

A simplification has been attempted in the procedures for determining the matrix elements of the generators of the unitary group U(n) over a tensor basis spanning the irreducible representation 2 ^N/2–S , 1 ^2S for an N-electron system. It has been shown that these matrix elements require, for their determination, only the corresponding representation matrices of cyclic permutations of the group S _N . A viable algorithm has been obtained for determining these representation matrices.     

Matrix‐covariant representation of high‐order configuration interaction and coupled cluster theories

We present the closed form of the reduced density matrices (RDMs) of arbitrary order for configuration interaction (CI) wave functions at any excitation level, up to the full CI. A special operator technique due to Bogoliubov is applied and extended. It focuses on constructions of matrix‐covariant expressions independent of the basis set used. The corresponding variational CI equations are given in an explicit form containing the matrices related to conventional excitation operators. A subsequent transformation of the latter to an irreducible form makes it possible to generate the matrix‐covariant representation for coupled cluster (CC) models. Here this transformation is performed for a simplified high‐order CC scheme somewhat reminiscent of the quadratic CI model. A generalized spin‐flip approximation closely related to high‐order CI and CC models is presented, stressing on a possible inclusion of nondynamical and dynamical correlation effects for multiple bond breaking. A derivation of the full CI and simple CC models for systems involving effective three‐electron interactions is also given, thereby demonstrating the capability of the proposed method to deal with complicated many‐body problem. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Algebraic solutions for the octahedral group: Group chain O⊃C4

Using double‐induced representation and eigenfunction method, algebraic expressions are derived for irreducible matrices, projection operators, and symmetry‐adapted functions in the group chain O⊃C₄ for both single‐valued and double‐valued representations. The simplicity of these expressions lies in the fact that they are functions of the quantum numbers of the corresponding group chain (the analogy of j, m for the group chain SO₃⊃SO₂) instead of the irreducible matrix elements. The symmetries of the symmetry‐adapted functions are disclosed. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 74: 7–22, 1999     

Theoretical investigation of Raman optical activity signatures of Tröger's base

The Raman and VROA spectra of (S,S)-Tr?ger's base are simulated. We mainly discuss the peaks in the 1140-1400 cm(-1) wavenumber range where an intense VROA signature is found. In this range, nearly all of the Raman-active bands belong to the irreducible representation A (C(2) point group), whereas no such observation is made for the VROA spectrum. The vibrational normal modes associated with the peaks in this range mainly consist of wagging and twisting motions of the hydrogen atoms. From the atomic contribution patterns (ACPs) and the group coupling matrices (GCMs), one finds that the VROA backward-scattering intensities mainly arise from hydrogen and carbon atoms in the vicinity of the two chiral nitrogen atoms. The VROA signatures in the 1140-1400 cm(-1) range are therefore a fingerprint of the local chirality around the two chiral nitrogen centers.     

Group theory and normal coordinate calculations

A method is proposed to obtain a set of computer generated symmetry adapted basis vectors which completely factorize a matrix associated with any operator which commutes with all the symmetry operations of a molecule. The cartesian coordinates of all atoms are used to derive the irreducible representations of the appropriate molecular point group and the transformation properties of any starting set of coordinates. Symmetry coordinates are then projected out each irreducible representation.     

Representations of the symmetric group generated by projected spin functions: A graphical approach

The representation matrices generated by the projected spin functions have some very interesting properties. All the matrix elements are integers and they are quite sparse. A very efficient algorithm is presented for the calculation of these representation matrices based on a graphical approach and a new indexing scheme for representation of primitive spin functions is introduced. Test calculations show that the method is very fast and suited for calculations on vector computers. © 1996 John Wiley & Sons, Inc.