Many-electron Functions Research Articles

Reduced Common Molecular Orbital Basis for Nonorthogonal Configuration Interaction.

Electron and charge transfers are part of many vital processes in nature and technology. Ab initio descriptions of these processes provide useful insights that can be utilized for applications. A combination of the embedded cluster material model and nonorthogonal configuration interaction (NOCI), in which the cluster wave functions are expanded in many-electron basis functions (MEBFs) consisting of spin-adapted, antisymmetrized products of multiconfigurational wave functions of fragments (which are usually molecules) in the cluster, appears to provide a compromise between accuracy and calculation time. Additional advantages of this NOCI–Fragments approach are the chemically convenient interpretation of the wave function in terms of molecular states, and the direct accessibility of electronic coupling between diabatic states to describe energy and electron transfer processes. Bottlenecks in this method are the large number of two-electron integrals that have to be handled for the calculation of an electronic coupling matrix element and the enormous number of matrix elements over determinant pairs that have to be evaluated for the calculation of one matrix element between the MEBFs. We show here how we created a reduced common molecular orbital basis that is utilized to significantly reduce the number of two-electron integrals that need to be handled. The results obtained with this basis do not show any loss of accuracy in relevant quantities like electronic couplings and vertical excitation energies. We also show a significant reduction in computation time without loss in accuracy when matrix elements over determinant pairs with small weights are neglected in the NOCI. These improvements in the methodology render NOCI–Fragments to be also applicable to treat clusters of larger molecular systems with larger atomic basis sets and larger active spaces, as the computation time becomes dependent on the number of occupied orbitals and less dependent on the size of the active space.

Determination of many-electron basis functions for a quantum Hall ground state using Schur polynomials

A method for determining the ground state of a planar interacting many-electron system in a magnetic field perpendicular to the plane is described. The ground state wave-function is expressed as a linear combination of a set of basis functions. Given only the flux and the number of electrons describing an incompressible state, we use the combinatorics of partitioning the flux among the electrons to derive the basis wave-functions as linear combinations of Schur polynomials. The procedure ensures that the basis wave-functions form representations of the angular momentum algebra. We exemplify the method by deriving the basis functions for the 52 quantum Hall state with a few particles. We find that one of the basis functions is precisely the Moore–Read Pfaffian wave function.

The calculation of rare-earth levels in layered cobaltates Rx/3CoO2 (x≦1)

Abstract We studied theoretically the crystal field and Zeeman split electronic levels of trivalent rare earths that are distributed over trigonal prismatic sites of the layered R x / 3 CoO 2 system (closely related to sodium cobaltate Na x CoO 2 ). The calculations were done in the whole basis of 4fn configurations (up to 3000 many-electron determinantal functions) for the ideal trigonal symmetry D 3 h , as well as for the reduced symmetry C2v that takes into account a more distant neighborhood of the R-sites. Detailed data on the doublet and singlet states for Pr, Nd, Sm, Tb and Dy are presented. The obtained g-factor and the van Vleck susceptibility tensor components are used for calculations of anisotropic magnetic susceptibilities and their temperature dependencies.

SDS: the ‘static–dynamic–static’ framework for strongly correlated electrons

A genetic ‘static–dynamic–static’ (SDS) framework is proposed for describing strongly correlated electrons. It permits both simple and sophisticated parameterizations of many-electron wave functions. One particularly simple realization amounts to constructing and diagonalizing the Hamiltonian matrix in the same number of many-electron basis functions in the primary (static), external (dynamic) and secondary (static) subspaces of the full Hilbert space. It combines the merits of both internally and externally contracted configuration interaction as well as intermediate Hamiltonian approaches. When the Hamiltonian matrix elements between the contracted external functions, with the coefficients determined by first order perturbation, are approximated as the diagonal elements of the zeroth-order Hamiltonian H 0, we obtain a multi-state multi-reference second-order perturbation theory (denoted as SDS-MS-MRPT2) that scales computationally with the fifth power of the molecular size. Depending on how H 0 is defined, various variants of SDS-MS-MRPT2 can be obtained. For simplicity, we here choose H 0 as a multipartitioned Moller–Plesset-like diagonal operator. Further combined with the string-based macroconfiguration technique, an efficient implementation of SDS-MS-MRPT2 is realized and tested for prototypical systems of variable near-degeneracies. The results reveal that SDS-MSMRPT2 can well describe not only standard benchmark systems but also problematic systems. Taking SDS-MSMRPT2 as a start, the accuracy may steadily be increased by relaxing the contraction of the external functions and/or iterating the diagonalization–perturbation–diagonalization procedure. As such, the SDS framework offers a very powerful scenario for handling strongly correlated systems.

The construction of symmetry-adapted functions in configuration interaction calculations for molecules with one principal axis of symmetry

A scheme is presented for the automatic generation on the computer of lists of many-electron functions of definite spin and symmetry, for use in configuration interaction calculations of the electronic wave functions of molecules having one principal axis of symmetry. All such linearly independent functions that can be obtained from a given basis set may be generated, or a limit may be put on the number of electron promotions, relative to a ground-state function, to be considered. The many-electron functions are expressed in terms of complex symmetry orbitals, but are generally real, and practically all computations may be performed without the use of complex arithmetic. The calculation of matrix elements of the Hamiltonian for these functions is discussed.

The structure and transformation properties of correlation functions for open shell states of molecules in non-orthogonal ao as well as in MO bases

A rigorous derivation of many-electron correlation functions for constructing the exact wave function for open shell molecules is given. These correlation functions are formally projections from the exact many-particle wave function and as such are particular realizations of applying Sinanoǧlu's Partial Orthogonalization Method. Correlation functions are derived for both an orthonormal set of (molecular) orbitals and a non-orthogonal set of (atomic) orbitals which provide alternative, but equivalent descriptions of a molecule in the independent particle picture. The transformations between the “molecular orbital” and “atomic orbital” correlation functions are considered. Simplifications of these relations for closed shell systems are presented; these in turn lead to new types of correlation functions for open shells. These new correlation functions are generalizations of those originally introduced by Silverstone and Sinanoǧlu and are susceptible to direct physical interpretation in terms of excitations from the independent particle “open shell” sea. The results given here, especially those on the non-orthogonal AO basis should be useful for semiempirical molecular correlation energy methods, as well as in CI studies.

Symmetry-adapted integrals over many-electron basis functions and operators

We consider integrals over symmetry-adapted basis functions that involve the coordinates of more than one electron. We focus on basis functions that can be written as products of one-electron functions and (say) a two-electron function. We show first that the two-electron parts of the basis functions can be absorbed into the operator, resulting in an integral over only one-electron basis functions, but a more complicated many-electron operator. We then prove a general formula for expressing such integrals in terms of symmetry-distinct integrals only.

Core ionization energies of atoms and ions calculated using the generalized Sturmian method

The generalized Sturmian method for solving the many-electron Schrodinger equation is reviewed. The method is illustrated with calculations of the core ionization energies of a series of atoms and ions. It is shown that when the “basis potential” is chosen to be the actual attractive potential of the nuclei in the system being studied, convergence is rapid, and a correlated solution can be obtained without the use of the self-consistent field approximation. Furthermore, when many-electron basis functions of this type are used, the kinetic energy term disappears from the secular equation, the nuclear attraction potential is diagonal, and the Slater exponents of the basis functions are automatically optimized. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 79: 1–7, 2000

Structural-dynamic approach and path analysis of the intramolecular electron transfer in the bridged complexes [(NH3)5Ru-py-(C2H2) n -py-Ru(NH3)5]5+ withn = 0,1,2,3

The electron interaction transfer paths in the mixed-valence complexes[(NH3)5Ru-py-(C2H2)n-py-Ru(NH3)5]5+ are analyzed in the framework of perturbation theory. A structural-dynamic approach to treatment of the intramolecular electron transfer is proposed. This approach is based on the solution of the nonstationary Schrodinger equation in a basis of many-electron functions describing separate structural sites of the electron-transport chain RyII→pyridyl→ C2H2)n→pyridyl→RuIII.

Translation symmetry for many-electron functions

We extend the treatment of translation symmetry, which is usually limited to orbitals (one-electron functions), to many-electron functions. The distinction between translational invariance and translation symmetry is analysed. We define an explicit translation operator for many-electron functions. The many-electron translation operators are related to the many-electron momentum operator, of which several forms are discussed.

Third-order complete active space self-consistent field based generalized Van Vleck perturbation theory

A state-selective generalized Van Vleck perturbation theory (GVVPT), correct through third order, for the determination of the electronic structure of polyatomic molecules is presented. The method adapts Kirtman's GVVPT for use with complete active space self-consistent field (CASSCF), and orthogonal complement, zeroth-order many-electron functions. The tenability of the method is demonstrated by calculations on several well-known “benchmark” molecules: the Be atom, using a (12s5p1d)/[7s3pld] basis set, double zeta (DZ) H 2O; 6-31GN NH 2; the singlet—triplet and singlet—singlet energy separations of DZP CH 2; the excited states of DZP CH + 2; and the dissociation of DZP + F 2.

Potential energy surface for highly symmetric structures described in the Hartree-Fock method by non-Roothaan type wave functions

The difficulties involved in the Hartree-Fock method of calculation of the force constants for structures described by non-Roothaan type wave functions were an- alyzed. It was shown on the example of the C4(D4h) molecule that the Hartree- Fock potential surface in the vicinity of the stationary points studied is con- tinuous and smooth only in defined cross-sections. The formally-correct calcu- lation of these cross-sections requires the use of a special procedure described in the present article. A comparison has been made of the results of the calcu- lation of the force constants obtained by the Hartree-Fock method (in the MINDO/3) approximation) and by allowing for the minimal configurational interaction. There exist in systems with a cubic (T d, O, Oh, T, T h) and tetragonal (D~h , C4v, D2d) symmetry with an open electronic shell t b (N = 2, 3, 4) and e 2, the so-called "non-Roothaan" type states, in the calculation of which by the Hartree-Fock method special measures must be taken for matching the symmetry of the molecular skeleton and many-electron functions of state (3, 4). These measures consist in taking into account in the SCF equations in an explicit form the unitary transformation matrix U with an accuracy matching that upon which the basis of the degenerate MO of an open shell has been determined. (Depending on U are the vectorial bond coefficients amn(U) and bmn (u) in the expressions for the energy func- tional and the Fock operations, and also the coefficients of the expansion of the function of state with respect to Slater determinants (3).) With decrease in the symmetry of the system (for example, during structural distortions) the necessity of taking into account the above features is only partially eliminated. An- alysis shows that there is a unique memory effect, at low symmetry distortions, the systems seemingly "remembering" that in the high symmetry configufation they possessed on non- Roothaan type wave function character, with the characteristic dependence of the latter on U. The result of this is that in the vicinity of a symmetric nuclear configuration the ex- pressions for the energy of the system and its wave function (for one and the same spectro- scopic state) are different and depend on the character of the distorting coordinate Q. The last fact is of considerable importance in calculating the physical characteristics, which are derivatives of a Hamiltonian with respect to internuclear distances, in particular the force constants. In the present work, an analysis is made of the main difficulties arisin Z in the calculations of force constants for highly symmetric structures described by non-Roothaan type wave functions. As an application, an examination is made of the problem of the calculation of the force constants for a C~ (D4h) molecule in its lowest singlet non-Roothaan type state iBi~ (electronic configuration eg2). This problem is of interest in

Electronic wave functions near coalescence points.

The behavior of many-electron wave functions near points where the positions of the some of the electrons coincide or where they coincide with positions of nuclei is investigated. A representation theorem for many-electron functions near such points is given extending previous results of Kato (cusp conditions) and Hoffmann-Ostenhof and Seiler. Furthermore, the influence of the Pauli principle on the local behavior of fermionic wave functions near many-particle coalescence points is studied.

Localization of electron group functions

AbstractThe requirements of variational freedom of electron group functions and their spatial separation are basic statements of the group functions method as they are necessary for taking into account the intragroup electron correlation while neglecting the intergroup one. But these requirements seem to be inconsistent with one another. This contradiction can be removed using the notion of antisymmetrical annulment of many‐electron functions introduced in the present work. The transformation of group functions (GF) by means of functions antisymmetrically annulling (ASA) other GF's is proposed that does not affect the whole system's wave function but can be used for localizing GF's. The problem of construction of a function \documentclass{article}\pagestyle{empty}\begin{document}$ \tilde \Phi $\end{document} ASA a given Φ can be reduced to solving a system of linear algebraic equations. A sufficient condition of existing of nontrivial \documentclass{article}\pagestyle{empty}\begin{document}$ \tilde \Phi $\end{document} is obtained.

Influence of electron correlation on the character of the splitting of degenerate states in metalloporphyrin dianions

The lowest singlet and triplet energy levels of metalloporphyrin dianions have been calculated according to the extended Hartree-Fock method in the π-electronic approximation with consideration of electron correlation. The calculations confirm the deviation from Hund's rule observed for these compounds. The possibility of the application of the method indicated to the construction of many-electron functions with the correct symmetry in the case of orbital degeneracy has been discussed.

An improved optical potential for many-electron green's function calculation: LiH and H 2O

An analysis of an approximate expression for the optical potential given by its functional derivative formulation, shows which are the diagrams left out at each order. A possible method which attempts to introduce the missing diagrams without losing the advantages provided by the functional derivative approach, is proposed. Numerical results are reported for LiH and H 2O.

Valence bond theory of narrow-band conductors

A diagrammatic valence bond (VB) theory is introduced for solids with one valence state per site, Coulomb intersite interactions, and nearest-neighbor electron transfers leading to a bandwidth 4 | t|. The VB method is applied to partly-filled regular segregated stacks of acceptors (A) and donors (D), which occur in all π-molecular organic conductors. The dimensions of the site representation for N e = γN electrons or holes on N > N e sites are found, each VB assignment is represented by a diagram, and the interconnection of these many-electron site functions due to electron transfers is also obtained diagrammatically. The resulting configuration interaction (CI) between all orbital assignments is solved exactly in subspaces of total spin, S. The complete thermodynamics of N e = 4 electrons on chains of N = 4, 5, 6, and 7 sites are found for the Hubbard choice of on-site repulsion, U. The VB method extends the computation of the susceptibility χ( T) of partly-filled Hubbard models to the previously inaccessible regime U ∼ 4| t| which, however, is suggested by experiment. Comparison with absolute χ( T) data for TTF-TCNQ is slightly improved for U ∼ 4| t| = 0.56 eV. Extensions of the VB method to more realistic potentials, to other data, and to N → ∞ extrapolations are discussed.

Proof of a fundamental theorem in the many-electron theory for atoms and molecules

In this work the cluster-function development for a many-electron function of Sinanoğlu's works is rigorously derived and generalized. More specifically, this development is given for the open shell cases, described by a function which contains internal correlation effects. The special case of closed shells is a very simple consequence of the more general formula obtained here.

Improved Quantum Theory of Many-Electron Systems. II. The Basic Method

A general method of obtaining accurate and useful many-electron wave functions for atoms and molecules is developed. The method involves the proper optimization of a many-electron function which is an eigen-function of total spin and which satisfies Pauli's principle. The procedure is somewhat similar to that in which one obtains the Hartree-Fock wave function by properly optimizing a Slater determinant. There are three key features of the new method: (1) The wave function remains accurate as the nuclei configuration is changed from the equilibrium configuration to that of the dissociated molecule; (2) the many-electron wave function can be interpreted in an independent-particle scheme; (3) the independent-particle states are no longer always required to be symmetry functions for the spatial symmetry group as in the Hartree-Fock method. In addition, the energy calculated using the new method is always lower than the Hartree-Fock energy, and the method is applicable to states of any total spin and number of particles. Calculations using this method are reported for the ${\mathrm{H}}_{2}$ and LiH molecules.

Atomic Many-Body Problem. I. General Theory of Correlated Wave Functions

The purpose of the paper is to present the general theory of atomic wave functions which explicitly depend on the distances between the electrons. The total wave function for the atom will be written as a composition of one-electron spin-orbitals, and 2-electron, 3-electron,... $N$-electron functions, respectively. The wave function has the following properties: (1) It becomes identical with the exact solution of the many-body problem if one expands the many-electron functions in terms of a complete set of Slater determinants; (2) it represents superposition of configurations if one writes the many-electron functions in the form of linear combination of a finite set of determinants; (3) one obtains the generalization of the Hylleraas ${r}_{12}$-method introducing the interelectronic distances explicitly into the many-electron functions.It will be shown that the many-electron functions can be orthogonalized with respect to the one-electron spin-orbitals, without restricting the generality of the total wave function. General formulas for the matrix components of the Hamiltonian with respect to correlated functions will be derived. The approximation containing only 2-electron correlations will be discussed, and it will be pointed out that in this approximation all matrix components can be reduced to 2-electron and 3-electron integrals, respectively; and the calculation of the 3-electron integrals may be simplified by introducing two interelectronic distances as integration variables.