Transferable integrals in a deformation density approach to crystal calculations. I. Crystal harmonics and their properties

The term “crystal harmonic” is introduced to denote a symmetrized plane wave in the special case where the wave vector is a reciprocal lattice vector. Crystal harmonics, thus defined, have the translational symmetry of the lattice, and they also have the transformation properties of the irreducible representations of the crystal's point group. An expansion is derived expressing crystal harmonics in terms of spherical Bessel functions and in terms of the functions ??_??,ξ (eigenfunctions of L² which are also basis functions for IRS of the crystal's point group). A sum rule for the functions ??_??,ξ is derived. Methods are given for expanding periodic functions of special symmetry in terms of crystal harmonics. Methods are also presented for calculating matrix elements of the potential in a crystal using crystal harmonics as a basis and for transforming to a STO basis. It is shown that the invariant component of the product of two crystal harmonics can be expressed as a sum of a few invariant crystal harmonics, and expressions for the coefficients in the sum are derived. Orthogonality with respect to summation over networks of points and normalization are also discussed. The properties mentioned above are illustrated in detail in the case of cubic crystals with point group O_h.     

The use of irreducible operators for determining the complete set of linearly independent crystal field parameters

A general method is given for finding the complete set of linearly independent crystal field parameters from symmetry arguments. No recourse is made to expansions of the crystal field in terms of spherical harmonics. The core of the method lies in an extension of the known zero-trace property of tensor operators, to the case of irreducible operators.     

Reference molecules for nonlinear optics: a joint experimental and theoretical investigation

Hyper-Rayleigh scattering (HRS) experiments and quantum chemical calculations are combined to investigate the second-order nonlinear optical responses of a series of reference molecules, namely, carbon tetrachloride, chloroform, trichloroacetonitrile, acetonitrile, and dichloromethane. The multipolar decomposition of the first hyperpolarizability tensor through the use of the spherical harmonics formalism is employed to highlight the impact of the symmetry of the molecular scatterers on their nonlinear optical responses. It is demonstrated that HRS is a technique of choice to probe the molecular symmetry of the compounds. Coupled-cluster calculations performed at the coupled-cluster level with singles, doubles, and perturbative triples in combination with highly extended basis sets and including environment effects by using the polarizable continuum model qualitatively reproduce the molecular first hyperpolarizabilities and depolarization ratios of the molecular scatterers.     

Numerical generation of hyperspherical harmonics for tetra-atomic systems

A numerical generation method of hyperspherical harmonics for tetra-atomic systems, in terms of row-orthonormal hyperspherical coordinates-a hyper-radius and eight angles-is presented. The nine-dimensional coordinate space is split into three three-dimensional spaces, the physical rotation, kinematic rotation, and kinematic invariant spaces. The eight-angle principal-axes-of-inertia hyperspherical harmonics are expanded in Wigner rotation matrices for the physical and kinematic rotation angles. The remaining two-angle harmonics defined in kinematic invariant space are expanded in a basis of trigonometric functions, and the diagonalization of the kinetic energy operator in this basis provides highly accurate harmonics. This trigonometric basis is chosen to provide a mathematically exact and finite expansion for the harmonics. Individually, each basis function does not satisfy appropriate boundary conditions at the poles of the kinetic energy operator; however, the numerically generated linear combination of these functions which constitutes the harmonic does. The size of this basis is minimized using the symmetries of the system, in particular, internal symmetries, involving different sets of coordinates in nine-dimensional space corresponding to the same physical configuration.     

Maxwell–Cartesian spherical harmonics in multipole potentials and atomic orbitals

 The nature of the Maxwell–Cartesian spherical harmonics S ⁽ⁿ⁾ _K and their relation to tesseral harmonics Y _nm is examined with the help of “tricorn arrays” that display the components of a totally symmetric Cartesian tensor of any rank in a systematic way. The arrays show the symmetries of the Maxwell–Cartesian harmonic tensors with respect to permutation of axes, the traceless properties of the tensors, the linearly independent subsets, the nonorthogonal subsets, and the subsets whose linear combinations produce the tesseral harmonics. The two families of harmonics are related by their connection with the gradients of 1/r, and explicit formulas for the transformation coefficients are derived. The rotational transformation of S ⁽ⁿ⁾ _K functions is described by a relatively simple Cartesian tensor method. The utility of the Maxwell–Cartesian harmonics in the theory of multipole potentials, where these functions originated in the work of Maxwell, is illustrated with some newer applications which employ a detracer exchange theorem and make use of the partial linear independence of the functions. The properties of atomic orbitals whose angular part is described by Maxwell–Cartesian harmonics are explored, including their angular momenta, adherence to an Uns?ld-type spherical symmetry relation, and potential for eliminating an angular momentum “contamination” problem in Cartesian Gaussian basis sets. Received: 9 July 2001 / Accepted: 7 September 2001 / Published online: 19 December 2001     

Fourier transform approach to potential harmonics

Because of the high degeneracy of hyperspherical harmonics, a method is needed for selecting the most important ones for inclusion in hyperangular basis sets. Such a method was developed by M. Fabre de la Ripelle, who showed that the most important harmonics are λ-projections of the product of the potential and a zeroth-order wave function; and he gave these the name, “potential harmonics.” In the present study we develop Fourier-transform-based methods for generating potential harmonics and for evaluating matrix elements between them. These methods are illustrated by a small calculation on three-body Coulomb systems with a variety of mass ratios. © 1997 John Wiley & Sons, Inc.     

Representation of cusps in a hyperspherical basis set

The wave functions of Coulomb systems have cusps at points corresponding to two particle coelescences. In this paper, we derive series representing the cusps in terms of hyperspherical harmonics multiplied by functions of the hyperradius. When the hyperspherical method is applied to Coulomb systems, the harmonics which appear in these series should be included in the hyperangular basis set.     

Equivalent spherical harmonics

It is known that the 2l+1 spherical harmonics Y_lm can be transformed into cyclically equivalent orbitals, of which only a few examples have so far been given explicitly. In this article the totality of such cyclic sets is derived. It is demonstrated that other kinds of equivalent spherical harmonics do not exist. Finally a set of five equivalent d orbitals related to icosahedral symmetry is introduced.     

A computational method for symmetry factorization without using symmetry operators

A computational method is suggested for symmetry factorization of symmetric matrices according to the symmetry properties of their basis. The method avoids the use of symmetry operators and is convenient for computer programming.     

Symmetry adaption reduced to tabulated quantities

The aim of this paper is to give an algebraic formula for symmetry-adapted linear combinations avoiding intuitive or laborious projection operator techniques. By utilization of the tabulated Clebsch-Gordan coefficients and surface harmonics of the point-groups the symmetry-adapted linear combinations are given in formula (4). A five-step algorithm is proposed and the example of a tetrahedron worked out. The relation to methods using site symmetry is discussed.     

铁磁性双核吲哚铜配合物的理论研究

采用密度泛函理论结合对称性破损态方法(DFT-BS),以双亚甲胺席夫碱为配体的双核吲哚铜配合物[Cu2L(AZ)(DMSO)](L=双甲亚胺三阴离子,由1-苯基-3-甲基-4-甲酰吡唑和1,3-二氨-2-丙醇衍生而成;AZ=7-氮杂吲哚阴离子)作为研究对象,通过与实验数据相比较,讨论了在不同密度泛函方法与基组对金属铜配物交换耦合常数的影响.结果表明,4种混合密度泛函方法(B3PW91,B3LYP,B3P86和PBE)及3种单一密度泛函(BPW91,BLYP和BP86)的计算结果都与实验值106cm-1有着相同的符号,但使用单一密度泛函所得到的结果与实验值比较接近.不论是混合密度泛函还是单一密度泛函计算所得到的耦合常数Jab对基组都有很大的依赖性,且BLYP/LANL2DZ水平下计算所得结果与实验数据吻合程度最好.研究表明,铁磁性配合物[Cu2L(AZ)(DMSO)]中存在着自旋离域和自旋极化效应.     

Optimization of molecular electronic structure calculations. 2. Calculation of symmetry-reduced matrix elements

A suggested formalism of the local symmetricized orbitals in conjunction with the selection technique for independent blocks of integrals in an original basis is used for a construction of multielectron Hamiltonian matrix elements in the symmetry orbital basis. The optimal molecular electronic structure calculation algorithm with the Hartree–Fock–Roothaan method in the symmetricized basis was obtained as a result. The minimal number of fundamentally distinguished (symmetry attributed) elements both in original and in symmetricized basis is used in the calculations.     

Symmetry contaminations in reactive scattering through long-lived collision complexes

We extend our Lanczos subspace time-independent wave packet method [J. Chem. Phys. 116 (2002) 2354] to investigate the issue of symmetry contaminations for the challenging deep-well H + O₂ reaction. Our central objective is to address the issue of whether significant symmetry contamination can occur if a wavepacket initially possessing the correct O–O exchange symmetry is propagated over tens of thousands of recursive steps using a basis which does not explicitly enforce the correct symmetry, and if so how seriously this affects the results. We find that symmetry contamination does exist where the symmetry constraint is not explicitly enforced in the basis. While it affects individual resonances and the associated peak amplitudes, the overall shape of the more averaged quantities such as total reaction probabilities and vibrational branching ratios are not seriously affected.     

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.     

Multielectron effects in high harmonic generation in N2 and benzene: simulation using a non-adiabatic quantum molecular dynamics approach for laser-molecule interactions

A mixed quantum-classical approach is introduced which allows the dynamical response of molecules driven far from equilibrium to be modeled. This method is applied to the interaction of molecules with intense, short-duration laser pulses. The electronic response of the molecule is described using time-dependent density functional theory (TDDFT) and the resulting Kohn-Sham equations are solved numerically using finite difference techniques in conjunction with local and global adaptations of an underlying grid in curvilinear coordinates. Using this approach, simulations can be carried out for a wide range of molecules and both all-electron and pseudopotential calculations are possible. The approach is applied to the study of high harmonic generation in N(2) and benzene using linearly polarized laser pulses and, to the best of our knowledge, the results for benzene represent the first TDDFT calculations of high harmonic generation in benzene using linearly polarized laser pulses. For N(2) an enhancement of the cut-off harmonics is observed whenever the laser polarization is aligned perpendicular to the molecular axis. This enhancement is attributed to the symmetry properties of the Kohn-Sham orbital that responds predominantly to the pulse. In benzene we predict that a suppression in the cut-off harmonics occurs whenever the laser polarization is aligned parallel to the molecular plane. We attribute this suppression to the symmetry-induced response of the highest-occupied molecular orbital.     

Multireference coupled-cluster study of the symmetry breaking in the C2B radical

The potential energy surfaces (PESs) for both the ground and the excited electronic states of the C(2)B radical are investigated using various multireference (MR) coupled-cluster (CC) approaches. In the ground state case we employ the reduced MR (RMR) CC approach with singles (S) and doubles (D), the RMR CCSD method, as well as its RMR CCSD(T) version corrected for secondary triples, relying on various model spaces and basis sets. The reliability of this approach is also tested against the benchmark full configuration interaction results obtained for a small Dunning-Hay (DH) basis set. The results imply a clear preference for a cyclic structure which, however, breaks the C(2v) symmetry. This symmetry breaking manifests itself strongly at the level of the independent particle model, as represented by the restricted open-shell Hartree-Fock approximation, but the tendency toward symmetry breaking diminishes with the increasing size of the basis set employed as well as with the enhanced account of the correlation effects. It is likely to disappear in the complete basis set limit. The general model space CCSD method is then used to compute vertical excitation energies for a number of excited states as well as the cuts of the PES as the boron atom moves around the C(2) fragment. These results also explain why no symmetry breaking is found when relying on a spin contaminated unrestricted Hartree-Fock reference, as in the UMP2 method.     

A numerical method to obtain a symmetry-adapted basis from the hamiltonian or a similar matrix

A practical method is proposed which using the hamiltonian matrix, or some other matrix corresponding to any operator with identical symmetry properties, enables one to obtain the transformation matrix, from the given basis to a symmetry-adapted basis. The method is very suitable for applications in the fields of molecular orbital and force constant calculations.     

A simple representation of energy matrix elements in terms of symmetry‐invariant bases

When a system under consideration has some symmetry, usually its Hamiltonian space can be parallel partitioned into a set of subspaces, which is invariant under symmetry operations. The bases that span these invariant subspaces are also invariant under the symmetry operations, and they are the symmetry‐invariant bases. A standard methodology is available to construct a series of generator functions (GFs) and corresponding symmetry‐adapted basis (SAB) functions from these symmetry‐invariant bases. Elements of the factorized Hamiltonian and overlap matrix can be expressed in terms of these SAB functions, and their simple representations can be deduced in terms of GFs. The application of this method to the Heisenberg spin Hamiltonian is demonstrated. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2010     

Lattice harmonics for graphite

A general theory for obtaining lattice harmonics of non-symmorphic space groups is presented, and the representation theory for these groups is briefly reviewed, with particular reference to graphite (space groupD _6h ⁴ [P6₃/mmc]). The irreducible matrix representations, compatibilities and lattice harmonics for alll are listed for all symmetry points, lines and planes in the representation domain of the Brillouin zone. The extra degeneracies introduced by time reversal are also considered. An appendix gives full details of the angular momentum conventions used in this paper.     

Incompletely symmetric non-bloch electron states in perfect cubic crystals. I. Eigenproblem and its solution

In the studies on impure metals where the perturbation potential due to the impurity has symmetry of the point group of the crystal, subdivision of the electron density of the pure metal into irreducible representations of the point group is of significance. A method is presented for the calculation of wave functions of the perfect cubic crystal which transform according to the incompletely symmetric irreducible representations of the point group. The functions were evaluated in the LCAO approximation using s-type AOS for the face centered cubic lattice. These are in the form of standing waves, and their coefficient functions are linear combinations of the products of the cubic harmonics of a suitable type and the spherical Bessel functions. Properties of the solutions obtained were examined. Numerical calculations were made for four irreducible representations of the point group.