On the use of symmetry in first-order perturbed Hartree–Fock theory

A method for the determination of the symmetry of first-order vectors in Hartree–Fock perturbation theory is developed. This leads to the definition of symmetry-adapted basis vectors to be employed at first order in the perturbation. It is shown that computer time can be saved, to some extent, in the calculation of second-order properties, by exploiting molecular symmetry. Specific examples are given for methane, ammonia, and water.     

New concepts in bonded functions theory II. Canonical set of high internal symmetry

A new definition of a canonical set of bonded functions has been introduced. For a given configuration with r singly occupied orbitals and given multiplicity, the set exhibits symmetry properties of a group of order r. It simplifies the evaluation of the energy matrix H as well as the orthogonalization procedure.     

The use of symmetry in the self-consistent field theory of the localised molecular orbital

It is shown how symmetry may be used in the context of the self-consistent field theory of the localised molecular orbital of the chemical bond. The results are (1) a reduction of the order of the secular determinant of the localised molecular orbital: (2) the classifying of the localised and canonical molecular orbitals under the various symmetry groups of the bonds and the molecule: (3) a clarification of the various types of symmetry group which are involved with the localised molecular orbitals.

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Zusammenfassung Es wird gezeigt, wie Symmetrieeigenschaften im Zusammenhang mit der SCF-Theorie lokalisierter Molekülorbitale benutzt werden kann. Die Ergebnisse sind: 1. eine Reduktion der Ordnung der Säkulardeterminante des lokalisierten Molekülorbitals; 2. die Klassifizierung der lokalisierten und kanonischen Molekülorbitale nach den verschiedenen Symmetriegruppen der Bindungen und des Moleküls; 3. eine Klärung der verschiedenen Typen von Symmetriegruppen, die im Zusammenhang mit lokalisierten Molekülorbitalen stehen.

     

Molecular symmetry. IV. The coupled perturbed Hartree–Fock method

Symmetry methods employed in the ab initio polyatomic program HONDO are extended to the coupled perturbed Hartree–Fock (CPHF) formalism, a key step in the analytical computation of energy first derivatives for configuration interaction (CI) wavefunctions, and energy second derivatives for Hartree–Fock (HF) wavefunctions. One possible computational strategy is to construct Fock-like matrices for each nuclear coordinate in which the one- and two-electron integrals of the usual Fock matrix are replaced by the integral first derivatives. “Skeleton” matrices are constructed from the unique blocks of electron-repulsion integral derivatives. The correct matrices are generated by applying a symmetrization operator. The analysis is valid for many wavefunctions, including closed- or open-shell spin-restricted and spin-unrestricted HF wavefunctions. To illustrate the method, we compare the computer time required for setting up the coupled perturbed HF equations for eclipsed ethane using D_3h symmetry point group and various subgroups of D_3h. Computational times are roughly inversely proportional to the order of the point group.     

Application of magnetically perturbed time-dependent density functional theory to magnetic circular dichroism. II. Calculation of A terms

The magnetically perturbed time-dependent density functional theory is used to derive equations for the magnetic circular dichroism (MCD) of degenerate transitions of closed shell molecules. The MCD of this type of transition can be divided into two contributions. The dominant contribution is usually that from A terms that arise because of the breaking of the degeneracy of the excited state in the presence of the magnetic field. The second contribution comes from B terms that arise because of the perturbation of the transition dipole by the magnetic field. The formalism is applied to ten tetrahedral d(0) transition metal oxy- and thioanions. The MCD parameters of these systems are reproduced quite well by the calculations. Simulated spectra derived from the calculated parameters are in good agreement with the observed spectra.     

A new theory for symmetry orbital and tensor (II)

The symmetry orbital-symmetry orbital tensor methtd is applied to the evaluation of molecular integrals (one-electron) and two-electron integrals) and the symmetry-orbital-tensor and self-consistent-field (SOT-SCF) calculations. A calculation sheme is proposed to simplify the evaluation of integrals and a key equation is derived to reduce the computation efforts in SCF iterations. According to the key equation, compared with the traditional SCF method, the computation efficiencies including CPU timing and external disk (or internal memory) requirement increase in the magnitude of the square of the order of a point group. The new SOT method is expected to be useful in the theoretical calculations of large molecular systems of high point group symmetries. Projcct supported by the National Natural Science Foundation of China (Grant No. 29473119).     

General formulation of the first-order perturbation graph theory

A generalization of the perturbation graph theory of Herndon and Párkányi is developed, enabling the calculation of resonance energies of conjugated heterocyclic and inorganic systems possessing any number of π-electrons.     

On the impact and the separation approximations in the theory of multiphoton interactions ivith thermally perturbed systems

Criteria are provided for the factorization of thermal averages incuring in the calculation of cross-sections for several-photon processes, into produc     

The use of site symmetry in constructing symmetry adapted functions

It is shown that the application of a projection operator from a given group to a function is equivalent to the successive application of projection operators from factor groups of the starting group to that function. When used with the factor groups representing the site symmetry of a position and the simplest group of interchanges of positions, this concept provides a very simple method for obtaining symmetry adapted linear combinations of basis functions.     

On the use of orientational restraints and symmetry corrections in alchemical free energy calculations

Alchemical free energy calculations are becoming a useful tool for calculating absolute binding free energies of small molecule ligands to proteins. Here, we find that the presence of multiple metastable ligand orientations can cause convergence problems when distance restraints alone are used. We demonstrate that the use of orientational restraints can greatly accelerate the convergence of these calculations. However, even with this acceleration, we find that sufficient sampling requires substantially longer simulations than are used in many published protocols. To further accelerate convergence, we introduce a new method of configuration space decomposition by orientation which reduces required simulation lengths by at least a factor of 5 in the cases examined. Our method is easily parallelizable, well suited for cases where a ligand cocrystal structure is not available, and can utilize initial orientations generated by docking packages.     

On the use of spatial symmetry in atomic-integral calculations: An efficient permutational approach

The minimal number of independent nonzero atomic integrals that occur over arbitrarily oriented basis orbitals of the form ?(r) · Y_lm(Ω) is theoretically derived. The corresponding method can be easily applied to any point group, including the molecular continuous groups C_∞v and D_∞h. On the basis of this (theoretical) lower bound, the efficiency of the permutational approach in generating sets of independent integrals is discussed. It is proved that lobe orbitals are always more efficient than the familiar Cartesian Gaussians, in the sense that GLO s provide the shortest integral lists. Moreover, it appears that the new axial GLO s often lead to a number of integrals, which is the theoretical lower bound previously defined. With AGLO s, the numbers of two-electron integrals to be computed, stored, and processed are divided by factors 2.9 (NH₃), 4.2 (C₅H₅), and 3.6 (C₆H₆) with reference to the corresponding CGTO s calculations. Remembering that in the permutational approach, atomic integrals are directly computed without any four-indice transformation, it appears that its utilization in connection with AGLO s provides one of the most powerful tools for treating symmetrical species.     

Nonlinear response theory with relaxation: the first-order hyperpolarizability

Based on the Ehrenfest theorem, an equation of motion that takes relaxation into account has been presented in wave-function theory, and the resulting response functions are nondivergent in the off-resonant as well as the resonant regions of optical frequencies. The derivation includes single- and multideterminant reference states. When applied to electric dipole properties, the response functions correspond to the phenomenological sum-over-states expressions of Orr and Ward [Mol. Phys. 20, 513 (1971)] for polarizabilities and hyperpolarizabilities of an isolated system. A universal dispersion formula is derived for the complex second-order response function. Response theory calculations are performed on lithium hydride and para-nitroaniline for off-resonant and resonant frequencies in the electro-optical Kerr effect and second-harmonic generation.     

Generalized perturbation theory: Quality of the first-order wave function

A recently proposed version of generalized perturbation theory, in which the whole energy correction is taken care of in second order, is investigated with respect to the quality of its first-order wave function. It is demonstrated that the overlap of the wave function generated in this procedure with the exact solution is in most cases much closer to unity than those of the Rayleigh-Schroedinger or Brillouin-Wigner perturbation theories. Certain approximations, by means of which realistic systems become amenable to investigation within the presently discussed framework, are studied.Based on a section of a thesis to be submitted by N. M. to the senate of the Technion — Israel Institute of Technology, in partial fulfilment of the requirements for the D.Sc. degree, and presented in the Second International Congress of Quantum Chemistry, New Orleans, April 1976.     

Longitudinal acoustic mode in polymers. II. Solution of the general composite perturbed elastic rod

The theory of the longitudinal vibrations of a composite rod with perturbing forces at the ends has been generalized by the inclusion of inertial effects at the ends of the rod. The frequencies have been calculated for both symmetric and asymmetric forces and masses. The problem of coupled rods has also been treated, and its relation to the isolated rod is shown. These results should be applicable to the study of longitudinal acoustic modes in long chain molecular systems.     

On the theory of electron correlation in atoms and molecules. II. General cluster expansion theory and the general correlated wave functions method

An exact cluster expansion of many electron wave functions is derived, beginning with a finite linear combination of Slater determinants rather than the more usual single determinant. This general cluster expansion is found to apply both in the case where all possible Slater determinants from a finite set of spin orbitals are included in the linear combination, and in the case where the number of determinants is restricted. The special properties of that finite linear combination of determinants closest to the exact wave function in the least squares sense are studied. These properties lead to the derivation of a general correlated wave functions method, illustrating again the close relationship between methods of this type and cluster expansion theory. Additional approximations, necessary for practical calculations, are set out.     

On the use of symmetry in the ab initio quantum mechanical simulation of nanotubes and related materials

Nanotubes can be characterized by a very high point symmetry, comparable or even larger than the one of the most symmetric crystalline systems (cubic, 48 point symmetry operators). For example, N = 2n rototranslation symmetry operators connect the atoms of the (n,0) nanotubes. This symmetry is fully exploited in the CRYSTAL code. As a result, ab initio quantum mechanical large basis set calculations of carbon nanotubes containing more than 150 atoms in the unit cell become very cheap, because the irreducible part of the unit cell reduces to two atoms only. The nanotube symmetry is exploited at three levels in the present implementation. First, for the automatic generation of the nanotube structure (and then of the input file for the SCF calculation) starting from a two‐dimensional structure (in the specific case, graphene). Second, the nanotube symmetry is used for the calculation of the mono‐ and bi‐electronic integrals that enter into the Fock (Kohn‐Sham) matrix definition. Only the irreducible wedge of the Fock matrix is computed, with a saving factor close to N. Finally, the symmetry is exploited for the diagonalization, where each irreducible representation is separately treated. When M atomic orbitals per carbon atom are used, the diagonalization computing time is close to Nt, where t is the time required for the diagonalization of each 2M × 2M matrix. The efficiency and accuracy of the computational scheme is documented. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2010     

The symmetry behaviour of the first-order density matrix and its natural orbitals for linear molecules

The general results regarding the symmetry of density matrices constructed from wave functions of a given symmetry species and the consequences for the transformation properties of the natural p-states reviewed in a previous communication [1] are illustrated for the special case p=1 for linear molecules with C _v - or D _h -symmetry. It is shown that even if the wave function belongs to a degenerate species, the natural orbitals (NO's) can always choosen to be adapted to the effective symmetry group C . The role played by the symmetry-adapted natural orbitals (SANO's) and of the natural expansion for 2-electron wave functions in this case is discussed.

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Zusammenfassung Die in einer früheren Arbeit [1] zusammengestellten Ergebnisse über das Symmetrieverhalten reduzierter Dichtematrizen, die aus Wellenfunktionen bestimmter Symmetrie konstruiert sind, sowie die Transformationseigenschaften der natürlichen p-Zustände werden am Beispiel von linearen Molekülen mit C _v - oder D _h -Symmetrie für den Fall p=1 illustriert. Es wird gezeigt, daß auch dann, wenn die Wellenfunktion zu einer entarteten Symmetriespecies gehört, die natürlichen Orbitale (NO's) der effectiven Symmetriegruppe C adaptiert sind.Die Rolle, die in diesem Fall die symmetrieadaptierten natürlichen Orbitale (SANO's) und die natürliche Entwicklung von 2-Elektronenwellenfunktionen spielen, werden diskutiert.

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Résumé Les résultats généraux recensés dans une communication précédente [1] concernant la symétrie des matrices densités construites à partir de fonctions d'onde de symétrie donnée et ses conséquences sur les propriétés de transformation des états -p naturels, sont illustrés sur le cas spécial où p=1 dans les molécules linéaires à symétrie C _v ou D _h . On montre que même si la fonction d'onde appartient à une catégorie dégénérée, les orbitales naturelles (NO's) peuvent toujours être choisies pour être adaptées au groupe de symétrie effectif C . On discute le rôle joué par les orbitales naturelles adaptées à la symétrie (SANO's) et l'expansion naturelle des fonctions d'onde à deux électrons.