Showing papers in "International Journal of Quantum Chemistry in 1973"

A “Level–Shifting” method for converging closed shell Hartree–Fock wave functions

Abstract: A discussion of Hartree damping in closed-shell Hartree-Fock theory is given indicating when the method might force convergence of intrinsically divergent cases. An alternative, but closely related principle, “level shifting” is described, and the advantages of the latter procedure discussed.

Half-Projected and Projected Hartree-Fock Calculations for Singlet Ground States. i. four-Electron Atomic Systems

Abstract: The half-projected Hartree-Fock function (HPHF) for singlet states is defined as a linear combination of two Slater determinants, which contains only spin eigenfunctions with even quantum number. Using a self-consistent procedure based on the generalized Brillouin's theorem, the RHF, HPHF and PHF functions are deduced for the ground states of the Li−, Be, B+, and C2+ systems, in a limited basis set. It is found that the HPHF function yields better energy values than the RHF function, very close to that of the PHF one. The HPHF scheme seems thus to be useful as a substitute for the PHF model, specially in the case of large electronic systems in which the latter method becomes unmanageable.

Hydrogen‐like relativistic corrections for electric and magnetic hyperfine integrals

Abstract: Expressions are derived for the n-dependent, hydrogen-like relativistic corrections to the magnetic dipole and electric quadrupole radial hyperfine integrals of an arbitrary atomic state. Good agreement has been obtained earlier between these corrections and the relativistic Hartree-Fock results for s1/2 states. The agreement remains good for the p1/2 states. For the p3/2 states in heavier atoms the hydrogenic corrections still reproduce the Hartree-Fock trend. For the higher l values the agreement is poor. Numerical tables are given for 1 ≤ Z ≤ 100 for the s1/2, p1/2, and p3/2 magnetic dipole cases and the p3/2 quadrupole case.

A configuration interaction study of the spin dipole-dipole parameters for formaldehyde and methylene

Abstract: The electron spin dipole-dipole contribution to the zero field splitting has been evaluated for the 3A2 (n π*) and 3A1 (π π*) states of formaldehyde using a CI wave function constructed from contracted Gaussian-lobe functions. The values D = 0.539 cm−1 and E = 0.031 cm−1 were obtained for the 3A2(n π*) state and D = −0.588 cm−1 and E = 0.058 cm−1 were obtained for the 3A1 (π π*) state using the CI wave function constructed from SCF orbitals of the respective parent configurations. An analysis of the effect of CI on the parameters is given for the 3A2 (nn π*) state of formaldehyde and the 3B1 ground state of methylene. Numerical results are given which show that internally consistent self-consistent field orbitals (ICSCF) are superior to canonical SCF orbitals as a starting point for a CI calculation. Our CI wave function for the 1A1 ground state gave an energy of −114.13658 hartrees which is significantly lower than any previously reported energy calculation. This wave function gives a dipole moment of 2.22 Debye (C+O−) in good agreement with the experimental value of 2.33 ± 0.02 Debye.

Use of molecular symmetry in SCF calculations

Abstract: A method is described whereby molecular symmetry properties may be used to reduce the numbers of one- and two-electron integrals that need to be calculated and stored in the course of a molecular SCF calculation. The method is a generalization of a previously reported procedure, extending the earlier work to cover those molecules belonging to point groups which have complex representations. The practical application of the method is discussed and an illustrative example given. The quite extensive tables of molecular symmetry properties which the method uses may be computer generated in a straightforward manner. A procedure for doing this using a minimum amount of input data is presented.

Spin projected extended Hartree--Fock equations

Abstract: The (spin projected) extended Hartree-Fock equations are derived and discussed in the general (“many λi”) case. The derivation is based on the corresponding generalized Brillouin theorem, substituting in its expression the spin projected DODS Slater determinant as a linear combination of determinants. The expressions obtained for the elements of the different matrices ecd (c = a or b, and d = a or b) occurring in the equations are analyzed. The transformation of the equations into the form of a pseudoeigenvalue problem and the LCAO form of the equations are given, too. Finally the relation to the Goddard's GF equations is discussed in detail.

Determination of one‐centre core integrals from the average energies of atomic configurations

Abstract: One-center core integrals for valence orbitals are determined from the experimental average energies of neutral atomic configurations from Li through Zn. These values are compared with those estimated from CNDO/1, “INDO/1”, CNDO/2, “INDO/2” and with theoretical values calculated from a pseudo-potential method. The agreement is good between values obtained from neutral atoms and from the psuedo-potential calculation except for the 3d orbitals of the transition elements where the theoretically calculated integrals over single ξ functions are not realistic. These two methods reproduce both term and average configuration energies for the first two rows of atoms; the semiempirical method reliably reproduces them for the third row. The CNDO/1 and INDO/1 methods underestimate atomic energies, while the CNDO/2 and INDO/2 procedures fail rather poorly. The propriety of using core integrals estimated semiempirically in molecular orbital calculations is discussed.

A study of the AMO method as applied to the lithium metal. I. Review, results, and discussion

Abstract: The total electronic energy per atom in the lithium metal has been calculated for three different Fermi surfaces as a function of the internuclear distance. In each case the optimized AMO energy as well as the total energy corresponding to doubly filled molecular orbitals (MO) has been calculated. For densities around the equilibrium density the spherical Fermi surface yields the lowest energy whereas a cubic Fermi surface is preferred for low densities. For densities around the equilibrium there is no band splitting: the AMO energy coincides with the MO energy. The computations have been carried out within an LCAO approximation with overlap and multicenter-integrals calculated accurately.

An Ab Initio calculation of the spin dipole‐dipole parameters for methylene

Abstract: The electron spin dipole-dipole interaction in CH2 has been calculated as a function of bond angle with configuration-interaction wave functions built from contracted gaussianlobe basis functions The values D = 0781 cm−1 and E = 0050 cm−1 were obtained for the spin dipole-dipole contribution to these parameters for the best CI wave function at the equilibrium geometry The angular dependence of D shows that the assumption of perfect orbital following is not valid Based on previous estimates of the spin-orbit contribution to D, the total D is estimated to be 09 ± 01 cm−1 which is higher than the current experimental value 076 ± 002 cm−1

Time‐dependent variational principle

Abstract: The time-dependent variational principle due to Frenkel is derived from Hamilton's principle using a suitable general expression for the variational wave function. The connections with a recent comment of P. O. Lowdin and P. K. Mukherjee on the same subject are discussed.

Lie algebras in quantum chemistry: Symmetrized orbitals

Abstract: The application of Lie algebras in quantum chemistry is considered. Particular attention is devoted to their application to high symmetry problems especially where icosahedral symmetry prevails. A general programme for implementing the theory of Lie algebras in the analysis of symmetry problems is outlined.

Modified Gaussian functions and their use in quantum chemistry. I. Integrals

Abstract: A modified Gaussian function g(u, v, w, a, R) = const s(a, R) is considered where l = u + v + w, s (a, R) is a 1s-type Gaussian function centered at R, a is the coefficient in the exponent of the 1 s Gaussian function and X, Y, Z are components of R. General formulae are derived for overlap integrals, kinetic energy integrals, nuclear attraction integrals, and electron repulsion integrals, valid for any l. The formulae are much simpler than those derived by Huzinaga for Cartesian Gaussian functions.

Stability conditions for the solutions of the hartree‐fock equations. VII. Stability of some closed‐shell nonalternant systems

Abstract: The instabilities of the solutions to the Hartree-Fock equations for the nonalternants in the pentalene, heptalene, etc., series and in the azulene, etc., series are examined. The systems are found to have symmetry-adapted solutions which are unstable for values of the core integral β close to the standard (spectroscopic) value; for example, the pentalene solution is unstable with β equal to 75% of its standard value. The “broken” symmetry solutions although exhibiting only a very slightly lower energy (0.01 eV) may exhibit dramatically different values for other properties, e.g., 30% changes in bond orders. The off-diagonal charge-density wave (CDW) appearing in the “broken” symmetry solutions at the onset of instability is amplified as the cooperative phenomena dominate, until in the “fully correlated” limit, the linked-ethylenic (bond alternating) structure is obtained.

Exact calculation of exchange polarization energy for H ion

Abstract: The exchange polarization energy and higher-order exchange corrections arising in the second order of the Murrell-Shaw and Musher-Amos (MS-MA) theory are evaluated analytically for the hydrogen molecule ion. At large internuclear distances the second-order exchange energy predicted by MS-MA theory behaves like . It represents only 80% of the difference between the exact result of Holstein and the asymptotic value of the first-order exchange energy. The same result holds for the case of Hirschfelder-van der Avoird (HAV) and Hirschfelder-Silbey (HS) theories.

Technique d'optimisation de l'energie SCF en fonction des exposants de slater

Abstract: We describe a method to optimize simultaneously Slater orbital exponents. The procedure is based on Newton's method and requires the derivatives of the energy as a function of the exponents. The calculation of these derivatives is described explicitly. The method has been applied to the hydrogen molecule.

Gauge invariant gaussian orbitals and the ab initio calculation of diamagnetic susceptibility for molecules

Abstract: It is shown that gauge terms can be introduced into the Gaussian functions used as the basis functions for an ab initio calculation of the energy of a molecule in the presence of a uniform magnetic field so that all the integrals become independent of the origin of the vector potential. The perturbation treatment of the diamagnetic susceptibility is considered in the molecular orbital approximation. The results show that the susceptibility can be calculated using only the unperturbed orbitals and their first-order corrections. All the integrals that arise can be expressed in terms of known functions.

Electrostatic representations of molecular groups

Abstract: The treatment of electrostatic interactions between molecular groups is based on quantum mechanical electron density distributions. A general method is suggested using tesseral harmonic multipole expansions and representing each set of point multipoles by a cluster of discrete charges small with respect to atomic dimensions. The procedure is carried out for the amino group with an electron density derived from the wave function of ammonia. We evaluate the optical rotation of two aminopyrrolidones as a test case. Expansions are compared for several approximate wave functions for ammonia.

CNDO/2 and MINDO/2 energy band structures of some homopolynucleotides

Abstract: The all-valence electron band structures of the periodic DNA models polyC, polyT, and polyA have been calculated with the aid of the CNDO/2 and MINDO/2 crystal orbital (CO) method. According to the obtained results the valence and conduction bands are always of π type. The widths of the bands are usually larger in the CNDO case, than those obtained for the same systems in the PPP CO approximation. On the other hand the MINDO results show similar widths as the PPP ones. The forbidden bandwidths are by 3–4 eV larger in the CNDO/2 case, than in the PPP one, while the MINDO/2 CO method has resulted in similar gaps like the PPP calculation.

Modified Cartesian Gaussian functions and their use in quantum chemistry

Abstract: The general formula for the electron-electron repulsion integral in the modified Cartesian Gaussian basis set derived in Ref. [1] is simplified. A general relation between the standard and modified CG functions is given. A possible use of the modified CG functions to quantum chemical calculations which include the correlation factor rij2 is indicated.

Convergence of the completely separated bipolar expansion of r 12−1

Abstract: A proof of the convergence of the double infinite series form of the radial factor in the bipolar expansion of r is given. It consists of demonstrating the validity of the required term-by-term integrations.

Matrix elements of general configuration of nonorthogonalized orbitals in state with definite spin

Abstract: The expressions for matrix elements of spin-independent operators F and G with the use of Young operators are obtained in the case of arbitrary nonorthogonalized orbital configuration, including both singly and doubly filled orbitals and corresponding to the definite value of total spin. The formulas obtained do not require a preliminary construction of linear combinations of determinants and turn into known Lowdin formulas in the particular case of fully antisymmetric spatial states.

Application of the configuration‐interaction method and the random phase approximation to the Ab Initio calculation of electronic excitation energies of H2o

Abstract: By performing calculations on H2 O similar to the calculations of Dunning and McKoy on C2H4 and HCHO [1a], it is shown that the singlet excited states of the water molecule cannot be adequately represented by an ab initio calculation (no semiempirical elements) using a valence-like basis set (either minimum basis or double zeta) of Slater-type orbitals. The triplets which are the lowest states of their symmetry appear to be described more accurately than the other states, and the lowest four triplet excitation energies are calculated to be 8.1, 8.4, 9.5, and 10.3 eV. The implications for the applicability of simple molecular orbital theory of the type commonly applied are discussed.

Sequence‐adapted molecular tensors: Algebraic methods and application to crystal field theory

Abstract: Tensorial sets adapted to sequences of finite subgroups are applied to the crystal field problem, and a general method for generating sequence-adapted molecular tensors using finite group algebra is formulated. All subgroup sequences of the abstract finite group G(24), isomorphic to the octahedral, O, tetrahedral, Td, and symmetric, S(4), groups are tabulated with explicit isomorphisms provided. The sequences fall into eight equivalence classes. A catalog of irreducible representations of G(24) adapted to a member of each of the eight sequence classes is given together with the transformations which generate representations adapted to all other sequences. With this data it is possible to systematically generate tensorial sets adapted to any sequence of a realization of G(24). Unitary transformations which adapt conventional forms of first- and second-rank irreducible tensorial sets of the rotation group to the eight sequences of the octahedral group are provided. Forms suitable for use with magnetic fields are included. The problem of a d1 ion in a trigonal crystal field is treated with sequence-adapted molecular tensors, and the utility of different sequences for descent in symmetry is discussed.

An Ab Initio study of the permanganate ion

Abstract: An ab initio self-consistent computation has been performed on the permanganate ion (MnO) in a large basis set of contracted gaussian functions. The ground state description as well as those of the first two excited states are discussed and compared with those obtained experimentally from photoelectron spectroscopy, ESCA, optical spectroscopy and magnetic circular dichroism.

Some developments in the spin‐extended Hartree‐Fock method

Abstract: All the second-order density matrix spin components for the spin-extended Hartree-Fock method are obtained. The coefficients in the final formulae are only ωsM, ωsM±1, ωsM±2, where ωsM are the weights of pure states of spin s in the initial unprojected determinant with spin projection M. The eigenvalue problem for the best electron density natural orbitals in the spin-extended method is formulated. All the second-order transition density matrix spin components between pure spin basis functions built of orthogonal orbitals and distinguished by different core choice are also found. This basis may be used on CI calculations.

An application of Green's functions to the study of the vibration-translation coupling of trapped oscillators in a linear chain

Abstract: The Green functions and Kubo linear-response theory are used to calculate the absorption coefficient of a collection of diatomic molecules embedded in a one-dimensional lattice. The effect of the environment is restricted to the coupling between the molecular internal vibration and lattice vibrations. For the molecular oscillators, both harmonic and anharmonic models are considered. The absorption line shape is expressed in closed form whether the molecular frequency falls outside or within the allowed phonon frequency range. In both cases a frequency shift is found. In the anharmonic case there is a broadening of the absorption lines. Expressions for the shift and line width are given.

Ab initio valence bond calculations. I. Methylene

Abstract: An ab initio valence bond (VB) method for calculating energies and wave functions for simple electronic systems is described and the results of its application to methylene are given together with extensive comparisons with previous theoretical results.

Direct minimization of the energy functional in LCAO–MO calculations

Abstract: In this paper we obtain formulae useful in methods for the direct minimization of the energy functional in the LCAO-MO-MC-SCF approach. The formulae are appropriate for dealing with variations in both the linear and nonlinear parameters. We include formulae for the usual closed- and open-shell problems as special cases.

Symmetry eigenfunctions suitable for many-electron theories and calculations. I. Mainly atoms

Abstract: An interpretation of degenenate symmetry spaces associated with a given atomic or molecular configuration is presented, and the theory necessary for atomic applications is discussed in detail. Lowdin's symmetric projection operator for spin and orbital angular momentum is applied on a well defined sequence of Slater determinants. In this way a degenerate is space is divided into orthogonal invariant subspaces which can be classified according to internal couplings of angular momentum, A procedure is found which permits expression of each of these LS eigenfunctions as the smallest possible number of successively orthogonalized projected Slater determinants. In particular, minimum size Hartree -Fock interacting spaces (important in rapidly convergent correlation energy calculations) are obtained in their most compact form. An optimum partition of degenerate spaces of three and four excited (atomic or molecular) configurations is obtained in genenal. Interpretations, simplifications, and systematizations in atomic configuration interaction calculations are discussed in light of present results. (auth)

A note on Löwdin orthogonalization and the square root of a positive self-adjoint matrix

Abstract: A quadratically convergent process using a continued fraction method for finding the square root of the overlap matrix used in Lowdin orthogonalization is presented. The continued fraction method is compared to several other methods for finding the square root of a matrix. The method is found to be more efficient than the other methods when high accuracy is desired.