Curvilinear and surficial electron holes in atoms and molecules

The antisymmetric property of many-electron wave functions results in the well-known Fermi hole, which implies that any two electrons with the same spin cannot be at the same point in space. We here point out that for certain types of antisymmetric wave functions, there exist curvilinear and surficial electron holes which imply that two electrons cannot be on particular curves and surfaces in space.     

A new look at correlations in atomic and molecular systems. I. Application of fermion monte carlo variational method

We are engaged in research directed toward the development of compact and accurate correlation functions for many-electron systems. Our computational tool is the variational method in which the many-electron integrals are calculated by Monte Carlo using the fermion Metropolis sampling algorithm. That is, a many-fermion system is simulated by sampling the square of a correlated antisymmetric wave function. The principal advantage of the method is that interelectronic distance r_ij may be included directly in the wave function without adding significant computational complexity. In addition, other quantities of physical and theoretical interest such as electron correlation functions and representations of Coulomb and Fermi “holes” are very easily obtained. Preliminary results are reported for He, H₂, and Li₂.     

Atomic wave function forms

Using variational Monte Carlo, we compare the features of 118 trial wave function forms for selected ground and excited states of helium, lithium, and beryllium in order to determine which characteristics give the most rapid convergence toward the exact nonrelativistic energy. We find that fully antisymmetric functions are more accurate than are those which use determinants, that exponential functions are more accurate than are linear function, and that the Padé function is anomalously accurate for the two-electron atom. We also find that the asymptotic and nodal behavior of the atomic wave function is best described by a minimal set of functions. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 63: 1001–1022, 1997     

Factored wave function for bound S-type states of two-electron atomic systems

A simple and accurate variational wave function in which the dependence in the interelectronic distance is factored is proposed to describe S-type states of two-electron atomic systems. We introduce a parameterization which generalizes the previous ones used in this same framework and which allows us to obtain in a simple way the wave function of both symmetric and antisymmetric excited states. We performed a systematic analysis of some exact properties such as the virial theorem and the cusp conditions and a study of both the one- and two-body densities. Finally, a comparison among the different correlation functions for these states was performed for helium. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 405–413, 1998     

Note on the space part of anti-symmetric wave functions in the many-electron problem

It is pointed out that if a many-electron antisymmetric wave function is expanded as a sum of spin-product functions, each multiplied by a function of coordinates, the resulting functions of coordinates have many of the same useful features found with the symmetric and antisymmetric functions representing singlet and triplet states in a two-electron system. For finding the energy, or any function of coordinates only, in the approximation in which spin-orbit interaction is neglected, one such function of coordinates can be used, the spins being disregarded. Simple procedures allow one to find matrix components of such operators as S ² and L . S from the functions of coordinates. These procedures are much easier to visualize than the use of projection operators, the permutation group, or other methods in current use. The general procedures are illustrated by application to the three-electron problem of the lithium atom, as treated by Lunell, Kaldor, and Harris, and their application to the contact hyperfine structure is pointed out.     

On the solution of the N-completeness problem

A solution is given for the problem of N-completeness which arises in practical calculations for many electron systems because of the necessity of employing a finite basis set of antisymmetric two-particle functions.     

Characterization of bond and current correlation structures in a molecular many-electron wave function

A method has been developed to analyzed the bond and current correlation structures of a molecular many-electron wave function. It is shown that the second order density matrix contains information about the bond and current correlations in its off-diagonal components with respect to the indices of orbital basis functions. We break down the off-diagonal correlation functions into five kinds: charge, spin scalar, spin quadrupole, charge spin, and spin polar correlation functions. For a real wave function, the four correlation functions, except for the spin polar one, have only symmetric–symmetric and antisymmetric–antisymmetric components. The former components give site–bond and bond–bond correlations of charges and spins, while the latter components give current–current correlations of charges and spins. The spin polar correlation function has only symmetric–antisymmetric components that give site–current and bond–current correlations of spins. The five off-diagonal correlation functions are expressed in terms of the off-diagonal components of the second order density matrix. The linked off-diagonal correlation functions are defined in that they give dynamical bond and current correlations. The method is applied to the analyses of the bond and current correlations in the low lying exact eigenstates of the PPP Hamiltonian of benzene.     

Shell model calculations: An efficient new algorithm

We describe an efficient new algorithm which extends the range of feasible shell model calculations. This algorithm is applicable to single shell and multiple shell configurations, where two or more quantum numbers (e.g., L and S) are required to label the states within each shell. The algorithm proceeds by factoring the shell model Hilbert space into a product of subspaces, one for each angular momentum. N-particle wave functions are built up recursively from N – 1 particle wave functions. Three kinds of N – 1- to N-particle coefficients are required to carry out the construction of N-particle electron (or fermion) states from N – 1 particle states. These are (1) coefficients of fractional parentage (CFP s) within a single shell, (2) outerproduct isoscalar factors (OISF s) within a single angular momentum subspace, and (3) innerproduct isoscalar factors (IISF s) which describe how multishell states within the complementary angular momentum subspaces are combined to form totally antisymmetric wave functions. All three types of N – 1- to N-particle coefficients are generated recursively using a single powerful and efficient matrix diagonalization algorithm. Matrix elements of single particle creation and annihilation operators are expressed in terms of single particle CFP s, OISF s, and IISF s. We also describe an efficient algorithm for computing matrix elements of products of creation and anihilation operators by inserting and summing over complete sets of intermediate states. This is the Feynman-like sum over path overlaps procedure. Timing benchmarks are presented comparing the new Drexel University shell model (DUSM ) code with a state of the art shell model code.     

Spectral density distribution of an N-electron Hamiltonian in a finite-dimensional and spin-adapted model space

Expressions for moments of spectral density distributions of a many-electron Hamiltonian defined in a finite-dimensional, antisymmetric, and spin-adapted model space (as, e.g., a full configuration interaction space) are derived. The moments are expressed in terms of combinations of two-electron integrals corresponding to a symmetric (a two-electron singlet) and antisymmetric (two-electron triplet) two-electron wave functions. A diagrammatic approach based on Hugenholtz-type diagrams and leading to a simple and universal classification scheme of the terms appearing in the expression for a specific moment is proposed. © 1995 John Wiley & Sons, Inc.     

The general scheme of using non-orthogonal radial orbitals in a complex electronic configuration of the atom

A theory for using non-orthogonal radial orbitals between shells with identical orbital quantum numbers, in the case of complex configurations, is presented. Construction of the antisymmetric wave function of the whole configuration, with the help of antisymmetrical wave functions of individual shells, is described. General methods of calculating matrix elements of one- and two-electron operators are given.     

Atomic spectral methods for molecular electronic structure calculations

Theoretical methods are reported for ab initio calculations of the adiabatic (Born-Oppenheimer) electronic wave functions and potential energy surfaces of molecules and other atomic aggregates. An outer product of complete sets of atomic eigenstates familiar from perturbation-theoretical treatments of long-range interactions is employed as a representational basis without prior enforcement of aggregate wave function antisymmetry. The nature and attributes of this atomic spectral-product basis are indicated, completeness proofs for representation of antisymmetric states provided, convergence of Schrodinger eigenstates in the basis established, and strategies for computational implemention of the theory described. A diabaticlike Hamiltonian matrix representative is obtained, which is additive in atomic-energy and pairwise-atomic interaction-energy matrices, providing a basis for molecular calculations in terms of the (Coulombic) interactions of the atomic constituents. The spectral-product basis is shown to contain the totally antisymmetric irreducible representation of the symmetric group of aggregate electron coordinate permutations once and only once, but to also span other (non-Pauli) symmetric group representations known to contain unphysical discrete states and associated continua in which the physically significant Schrodinger eigenstates are generally embedded. These unphysical representations are avoided by isolating the physical block of the Hamiltonian matrix with a unitary transformation obtained from the metric matrix of the explicitly antisymmetrized spectral-product basis. A formal proof of convergence is given in the limit of spectral closure to wave functions and energy surfaces obtained employing conventional prior antisymmetrization, but determined without repeated calculations of Hamiltonian matrix elements as integrals over explicitly antisymmetric aggregate basis states. Computational implementations of the theory employ efficient recursive methods which avoid explicit construction the metric matrix and do not require storage of the full Hamiltonian matrix to isolate the antisymmetric subspace of the spectral-product representation. Calculations of the lowest-lying singlet and triplet electronic states of the covalent electron pair bond (H(2)) illustrate the various theorems devised and demonstrate the degree of convergence achieved to values obtained employing conventional prior antisymmetrization. Concluding remarks place the atomic spectral-product development in the context of currently employed approaches for ab initio construction of adiabatic electronic eigenfunctions and potential energy surfaces, provide comparisons with earlier related approaches, and indicate prospects for more general applications of the method.     

Configuration interaction matrix elements. I. Algebraic approach to the relationship between unitary group generators and permutations

Matrix elements of unitary group generators between spin-adapted antisymmetric states are shown to be proportional to spin matrix elements of so-called “line-up” permutations. The proportionality factor is given explicitly as a simple function of the orbital occupation numbers. If one bases the theory on ordered orbital products, the line-up permutations are given a priori. The final formulas have a very simple structure; this is a direct consequence of the fact that the spin functions have been taken to be geminally antisymmetric.     

On the formulation of spin-free quantum chemistry

A set of spin-free functions ?_i(r),i = 1 …? f, is obtained which form the basis of spin-free quantum chemistry. The ?_i(r) show a one-to-one correspondence to antisymmetric space-spin functions Ψ_i(r, σ) with spin functions constructed according to Löwdin's projector operator method.     

Improved Hylleraas calculations for ground state energies of lithium ISO–electronic sequence

The ²S ground state of lithium iso–electronic sequence is calculated by the use of Hylleraas-type wave functions. A 92 term one-spin wave function was used for lithium atom calculations. The energy obtained was ?7.478031 a.u. as compared with the previous best value of ?7.478025 a.u. calculated by Larsson. In addition, improved energies for Z = 4 to 8 were calculated by the use of 60 term wave functions. This work thus provides the lowest ab initio ground state energies for lithium sequence to date.     

The application of strictly localized geminals to the description of chemical bonds

A theory is developed in which closed-shell molecules are viewed as systems of weakly interacting chemical bonds. Composite-particle creation operators obtained by an appropriate quasiparticle transformation are used to create the wave function of two-electron bonds. These quasiparticles are bosons, since they are composed of two electrons, but the total many-electron wave function is properly antisymmetric. The internal structure of the quasi-Bose-particles is affected by inductive interbond interactions. Delocalization and dispersion interactions between different bonds are neglected, thus the approach corresponds to a first-order many-body PT (Perturbation Theory) with a correlated, but fully localized, reference state. The whole formalism is developed ab initio. The nonorthogonality problem is handled by a biorthogonal formulation. To illustrate the effectiveness of the model, numerical calculations are reported.     

A scheme of numerical solution for three‐dimensional isoelectronic series of hydrogen atom using one‐dimensional basis functions

The solution of three‐dimensional Schrödinger wave equations of the hydrogen atoms and their isoelectronic ions (Z = 1 − 4) are obtained from the linear combination of one‐dimensional hydrogen wave functions. The use of one‐dimensional basis functions facilitates easy numerical integrations. An iteration technique is used to obtain accurate wave functions and energy levels. The obtained ground state energy level for the hydrogen atom converges stably to −0.498 a.u. The result shows that the novel approach is efficient for the three‐dimensional solution of the wave equation, extendable to the numerical solution of general many‐body problems, as has been demonstrated in this work with hydrogen anion.     

Product of group-function expansions for correlated wave functions

A theorem is proved which demonstrates the relationship between a product of group functions describing the correlated motion of a particular group of electrons in an N-electron system and a wave function obtained from the exact wave function which describes the correlation of the same group of electrons. By considering such products of group functions as elements in a variational wave function, an expansion for correlated wave functions is suggested, which emphasizes the correlated motion of groups of electrons in the whole system.     

New iterative scheme for a simultaneous calculation of m first eigenstates of a real symmetric matrix

A new iterative scheme for a simultaneous calculation of the m lowest eigenvalues together with their eigenvectors has been derived for a real symmetric matrix. The scheme is based on the orthogonal gradient method and is easy to use for large-scale configuration-interaction calculations of electronic wave functions. A variant of the scheme deals with nonorthogonal basis functions, which are particularly simple in the case of the bonded-function method of Boys.     

Configuration interaction matrix elements. II. Graphical approach to the relationship between unitary group generators and permutations

The explicit expressions for the matrix elements of unitary group generators between geminally antisymmetric spin-adapted N-electron configurations in terms of the orbital occupancies and spin factors, given as spin function matrix elements of appropriate orbital permutations, are derived using the many-body time-independent diagrammatic techniques. It is also shown how this approach can be conveniently combined with graphical methods of spin algebras to obtain explicit expressions for the spin factors, once a definite coupling scheme is chosen. This method yields explicit expressions for the orbital permutations defining the spin factors. However, if desired, the explicit determination of line-up permutations can be avoided in this approach, since they are implicitly contained in the orbital diagrams. It also clearly indicates why the geminally antisymmetric spin functions have to be used when a simple formalism is desired.     

Variational treatment on the energy of the He‐sequence ground state with weakest bound electron potential model theory

The concept of spin–orbital of the weakest bound electron is described used to construct the antisymmetric wave function of atomic or ionic systems within weakest bound electron potential model theory (WBEPM theory). The total energies of He‐sequence (Z = 2–9) in the ground states is calculated with a variational method. The effect of fixed orbital approximation is discussed quantitatively. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005