Perturbation improvement of SCF orbitals

A simple method for the perturbation improvement of SCF orbitals, based on the appropriate variation principle, is described. The method allows for the optimal extension of the original basis set without re-optimization of all the orbital exponents and guarantees the best lowering of the total energy. The numerical illustration of the proposed variation scheme is also given.     

A technique for orbital exponent optimization in ab initio HF?SCF?LCAO?MO calculations

A technique for Slater orbital exponent optimization in an HF? SCF? LCAO? MO calculation is proposed in which orbital exponent variation is incorporated into the SCF scheme. This is accomplished by rewriting Slater's rules so that the shielding terms depend on the molecular charge distribution through the elements of the population matrix. The SCF scheme then includes a calculation of a new set of orbital exponents from the coefficients of self-consistent molecular orbitals obtained from the previous set of exponents. The process is iterated until the energy attains its lowest value. The technique is illustrated by minimal basis calculations on LiH, BH, and HF. Near optimization is obtained with considerably less effort than is necessary for other reported techniques. Aside from interesting properties, the technique can be important for extended basis calculations where exponent optimization is a difficult task.     

Orbital correlation diagrams based on multiconfigurational variation of moments II. Application

A formalism was developed in the multiconfigurational variation of moments (MCM ) framework, which yields physically meaningful orbital energies for occupied and virtual orbitals starting from self-consistent field (SCF ) calculations. This is possible through a skillful distribution of the correlation energy on the orbital energies. The application of this method is demonstrated by SINDO 1 calculations on the dissociation of H₂ and the following symmetry-forbidden reactions: (1) torsion of ethylene; (2) ring opening of (a) cyclobutene, and (b) cyclopropyl cation; (3) cycloreversion of 1, 1-dicyano-2-methoxycyclobutane. The allowed reactions corresponding to 2a and 2b are investigated in the SCF scheme. The energy hypersurfaces are calculated for all reactions and the MO correlation diagrams are presented and discussed.     

Computation scheme for optimizing multiconfigurational wave-functions

A computation scheme is proposed to determine the wave-functions of molecular systems within the framework of the CMC SCF theory and the APSG SCF approach. The orbital optimization is carried out by the refined first-order one-electron Hamiltonian method. Explicit expressions of the first and second energy derivatives are obtained. In the suggested scheme all the calculations are based on using the matrices of the “partial” Coulomb and the exchange operators constructed over the orbitals at the current iteration cycle.     

Orbital relaxation in the Rydberg series of the lithium atom. An excited state SCF calculation

Self-consistent-field (SCF ) calculations for a series of Rydberg states (1s²ns)²S of the Li atom are performed using the generalized Brillouin theorem (GBT) method. The calculated energy is a proper upper bound to the excited state energy. The SCF term values of the Rydberg states are almost the same as those of the frozen-core approximation ones. The orbital behavior shows that the core is slightly expanded by the penetration of the Rydberg orbitals, and the higher Rydberg orbitals can be very well represented by the modified hydrogen-like orbitals.     

Self‐consistent shielding in atoms and molecules

The orbital exponents of trial wave functions for simple systems can be found from the potential energy terms alone. Shielding of the nuclear charge by one electron on another is determined by the relative values of the nuclear–electron attraction and the electron–electron repulsion. For two electrons in the same orbital, the shielding is divided equally. For different orbitals, only the inner electron shields the outer. The systems tested are first‐row atoms, using Slater orbitals. It appears that if this approach can be generalized, it may not be necessary to calculate kinetic energies in chemical systems, since they will be determined by the orbital exponents. This would be useful if trial wave functions were not available, but trial electron density functions were. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Relativistic corrections in the LCGTO–local-spin-density method. I. Atomic bases for transition metals

Compact orbital GTO basis sets optimized with a nonrelativistic Hamiltonian, then decontracted, are utilized in an SCF treatment with a quasi-relativistic scalar Hamiltonian including the mass–velocity and one-electron Darwin operators. Ionization and excitation energies, orbital energies, and radial mean values obtained from different expansion patterns have been tested in atomic calculations for Ag and generalized for Cu and Au atoms. The one-electron spin–orbit operator is also used in an SCF treatment. Spin–orbit coupling energies are calculated for Cu, Ag, and Au atoms.     

Atomic orbital basis sets for use with effective core potentials

Basis sets developed for use with effective core potentials describe pseudo‐orbitals rather than orbitals. The primitive Gaussian functions and the contraction coefficients in the basis set must therefore both describe the valence region effectively and allow the pseudo‐orbital to be small in the core region. The latter is particularly difficult using 1s primitive functions, which have their maxima at the nucleus. Several methods of choosing contraction coefficients are tried, and it is found that natural orbitals give the best results. The number and optimization of primitive functions are done following Dunning's correlation‐consistent procedure. Optimization of orbital exponents for larger atoms frequently results in coalescence of adjacent exponents; use of orbitals with higher principal quantum number is one alternative. Actinide atoms or ions provide the most difficult cases in that basis sets must be optimized for valence shells of different radial size simultaneously considering correlation energy and spin‐orbit energy. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 516–520, 2000     

Independent particle theory with electron correlation

We formulate an effective independent particle model where the effective Hamiltonian is composed of the Fock operator and a correlation potential. Within the model the kinetic energy and the exchange energy can be expressed exactly leaving the correlation energy functional as the remaining unknown. Our efforts concentrate on finding a correlation potential such that exact ionization potentials and electron affinities can be reproduced as orbital energies. The equation-of-motion coupled-cluster approach enables us to define an effective Hamiltonian from which a correlation potential can be extracted. We also make the connection to electron propagator theory. The disadvantage of the latter is the inherit energy dependence of the potential resulting in a different Hamiltonian for each orbital. Alternatively, the Fock space coupled-cluster approach employs an effective Hamiltonian which is energy independent and universal for all orbitals. A correlation potential is extracted which yields the exact ionization potentials and electron affinities and a set of associated molecular orbitals. We also describe the close relationship to Brueckner theory.     

Molecular states of atoms. II

Orbital energy parameters, previously obtained from atomic valence state energies, are used in calculating approximate wave functions for their orbitals. The radial factors of these wave functions are expressed as linear combinations of three Gaussian type orbitals with selected exponents, the coefficients being determined by normalisation and reproduction of the kinetic energy and interelectron repulsion parameters. Wave functions of universal form are obtained for the non-transition elements up to xenon. Each calculated s orbital wave function (except 1s) has a radial node, as is appropriate if there is a p orbital in the same shell with none.     

Molecule intrinsic minimal basis sets. I. Exact resolution of ab initio optimized molecular orbitals in terms of deformed atomic minimal-basis orbitals

A method is presented for expressing the occupied self-consistent-field (SCF) orbitals of a molecule exactly in terms of chemically deformed atomic minimal-basis-set orbitals that deviate as little as possible from free-atom SCF minimal-basis orbitals. The molecular orbitals referred to are the exact SCF orbitals, the free-atom orbitals referred to are the exact atomic SCF orbitals, and the formulation of the deformed "quasiatomic minimal-basis-sets" is independent of the calculational atomic orbital basis used. The resulting resolution of molecular orbitals in terms of quasiatomic minimal basis set orbitals is therefore intrinsic to the exact molecular wave functions. The deformations are analyzed in terms of interatomic contributions. The Mulliken population analysis is formulated in terms of the quasiatomic minimal-basis orbitals. In the virtual SCF orbital space the method leads to a quantitative ab initio formulation of the qualitative model of virtual valence orbitals, which are useful for calculating electron correlation and the interpretation of reactions. The method is applicable to Kohn-Sham density functional theory orbitals and is easily generalized to valence MCSCF orbitals.     

Basis set generation for the SCF calculation

A method for basis set generation for SCF calculations is proposed. Using SCF orbitals and orbital energies obtained in the extended basis set the Fock operator can be expressed as its spectral resolution. The sum of differences between occupied orbital energies and corresponding eigenvalues obtained by the diagonalization of this operator in the new smaller basis set is a criterion of the quality of this new set. The present method consists of the minimization of this sum by changing the parameters that determine the new basis functions. An example of the optimization of the different Gaussian basis sets for the LiH molecule is described.     

Determination de bases de gaussiennes optimales pour les molecules

Bases composed of one fixed linear combination ofn = 2 and 3 Gaussian functions have been investigated for the hydrogen molecule. It is shown that the scaling factor applied to Huzinaga's free atom exponents,which minimizes the total SCF energy, is independent of n and has the same value as forn = 1. This holds for every interatomic distanceD. From the energy variation as a function ofD, the force constant, the equilibrium distance and the bonding energy are deduced, for the different bases investigated, and compared with experimental values, and with values obtained by other authors by means of minimal bases of Slater orbitals.     

Correlated one-particle method: numerical results

In a previous paper a correlated one-particle method was formulated, where the effective Hamiltonian was composed of the Fock operator and a correlation potential. The objective was to define a correlated one-particle theory that would give all properties that can be obtained from a one-particle theory. The Fock-space coupled-cluster method was used to construct the infinite-order correlation potential, which yields correct ionization potentials (IP's) and electron affinities (EA's) as the negative of the eigenvalues. The model, however, was largely independent of orbital choice. To exploit the degree of freedom of improving the orbitals, the Brillouin-Brueckner condition is imposed, which leads to an effective Brueckner Hamiltonian. To assess its numerical properties, the effective Brueckner Hamiltonian is approximated through second order in perturbation. Its eigenvalues are the negative of IP's and EA's correct through second order, and its eigenfunctions are second-order Brueckner orbitals. We also give expressions for its energy and density matrix. Different partitioning schemes of the Hamiltonian are used and the intruder state problem is discussed. The results for ionization potentials, electron affinities, dipole moments, energies, and potential curves are given for some sample molecules.     

Towards a “Chemical” Hamiltonian

An analysis of the LCAO Hamiltonian is performed in terms of a “mixed” formulation of the second quantization for nonorthogonal orbitals, compressing the different interactions to one- and two-center terms as far as possible by performing appropriate projections. For this purpose an operator of atomic charge is also introduced, the expectation values of which are the Mulliken gross atomic populations on the individual atoms. The LCAO Hamiltonian is decomposed into terms having different physical meaning and significance: (i) sum of effective atomic Hamiltonians; (ii) the electrostatic interactions in the point-charge approximation; (iii) the electrostatic effects connected with the deviation of the actual charge distribution from the pointlike one; (iv) two-center overlap effects; (v) finite basis (“counterpoise”) correction terms related to the individual atoms; and (vi) similar finite basis correction terms with respect to the two-center interactions. Only terms of types (i) to (iv), containing no three- or four-center integrals, are considered as having physical significance. Based on the analysis of the Hamiltonian, an energy partitioning scheme is developed, and explicit expressions are given for one- and two-center (and basis extension) components of the SCF energy. The approach is also applied to the problem of intermolecular interactions, and an explicit formula is given permitting calculation of the “counterpoise” part of the supermolecule energy by properly taking into account that it depends not only on the extension of the basis, but also on the occupation of the additional orbitals in the intervening molecule—a factor completely overlooked in the usual scheme of calculations.     

Molecular states of atoms. I. Orbital energy parameters

Atomic valence state energies are analyzed to obtain values of orbital energy parameters that may be used in semiempirical molecular orbital calculations. Difficulty in defining the interaction between orbitals with non-integer electron populations is systematically avoided by distinguishing between a valence state and a molecular state of an atom, only the latter state having non-integer spin paired orbital occupancy. Application of the virial theorem to the molecular state enables a value for the orbital kinetic energy to be obtained from the valence state orbital energy parameters once an arbitrary configuration is defined as reference. The orbitals then are eigenfunctions of the atomic Fock operator for that reference molecular state and, with their energy parameters, may be employed as a fixed basis set for molecular orbital calculations.     

Hylleraas method for many‐electron atoms. I. The Hamiltonian

A general expression for the nonrelativistic Hamiltonian for n‐electron atoms with the fixed nucleus approximation is derived in a straightforward manner using the chain rule. The kinetic energy part is transformed into the mutually independent distance coordinates r_i, r_ij, and the polar angles θ_i, and φ_i. This form of the Hamiltonian is very appropriate for calculating integrals using Slater orbitals, not only of states of S symmetry, but also of states with higher angular momentum, as P states. As a first step in a study of the Hylleraas method for five‐electron systems, variational calculations on the ²P ground state of boron atom are performed without any interelectronic distance. The orbital exponents are optimized. The single‐term reference wave function leads to an energy of ?24.498369 atomic units (a.u.) with a virial factor of η = 2.0000000009, which coincides with the Hartree–Fock energy ?24.498369 a.u. A 150‐term wave function expansion leads to an energy of ?24.541246 a.u., with a factor of η = 1.9999999912, which represents 28% of the correlation energy. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Formulation of the coupled cluster theory with localized orbitals in correlation calculations on polymers

It is shown that in the framework of coupled cluster theory both the correlation energy per unit cell and quasi-particle band structures of polymers can be computed directly from matrix elements of the excitation operator and the two-electron integrals calculated in localized orbital basis. Further, it is described how to thake advantage of the localized nature of the orbitals applied. Ab initio test calculations on a finite model system similar to the Pariser–Parr–Pople Hamiltonian are presented.     

Improved convergence of self-consistence procedures in the MC SCF theory

To obtain optimized orbitals within the MC SCF theory, the energy surface near a chosen point is approximated by a quadratic function of independent matrix elements of a small orthogonal orbital transformation. The method of a second-order one-electron Hamiltonian (OEH) is developed on the basis of this approximation. A procedure is proposed to define step coordinates, insuring a rapid descent along an average-energy surface also in the cases when the matrix of second energy derivatives has eigenvalues negative or close to zero. The results obtained in applying the OEH method for the calculation of ground and triplet states of uracile in the π-electron approximation are discussed. When a complete matrix of the second energy derivatives is used, the self-consistence procedure is quadratically convergent. An exponential, yet rapid enough convergence is provided by a simplified computation scheme neglecting cross derivatives.