Fock expansion application

Of the many quantum chemical approaches available, density-functional theory (DFT) has over the past decade become a key method, with applications ranging from interstellar space, to the atmosphere, the biosphere and the solid state. The strength of the method is that whereas conventional ah initio theory includes electron correlation by use of a perturbation series expansion, or increasing orders of excited state configurations added to zero-order Hartree-Fock solutions, DFT methods inherently contain a large fraction of the electron correlation already from the start, via the so-called exchange-correlation junctional. 

Traditional hydrogenic orbitals used in atomic and molecular physics as expansion bases belong to the nlm) representation, which in configuration space corresponds to separation in polar coordinates, and in momentum space to a separation in spherical coordinates on the (Fock s) hypersphere [1], The tilm) basis will be called spherical in the following. Stark states npm) have also been used for atoms in fields and correspond to separation in parabolic coordinates an ordinary space and in cylindrical coordinates on (for their use for expanding molecular orbitals see ref. [2]). A third basis, to be termed Zeeman states and denoted nXm) has been introduced more recently by Labarthe [3] and has found increasing applications [4]. 

Establishing a hierarchy of rapidly converging, generally applicable, systematic approximations of exact electronic wave functions is the holy grail of electronic structure theory [1]. The basis of these approximations is the Hartree-Fock (HF) method, which defines a simple noncorrelated reference wave function consisting of a single Slater determinant (an antisymmetrized product of orbitals). To introduce electron correlation into the description, the wave function is expanded as a combination of the reference and excited Slater determinants obtained by promotion of one, two, or more electrons into vacant virtual orbitals. The approximate wave functions thus defined are characterized by the manner of the expansion (linear, nonlinear), the maximum excitation rank, and by the size of one-electron basis used to represent the orbitals. 

The l/REP(r), U ARKP(r), and terms At/f EP(r) in 11s0 of Eqs. (23), (31), and (34) or Eq. (6), respectively, are derived in the form of numerical functions consistent with the large components of Dirac spinors as calculated using the Dirac-Fock program of Desclaux (27). These operators have been used in their numerical form in applications to diatomic systems where basis sets of Slater-type functions are employed (39,42,43). It is often more convenient to represent the operators as expansions in exponential or Gaussian functions (32). The general form of an expansion involving M terms is... 

Fundamental to almost all applications of quantum mechanics to molecules is the use of a finite basis set. Such an approach leads to computational problems which are well suited to vectoris-ation. For example, by using a basis set the integro-differential Hartree-Fock equations become a set of algebraic equations for the expansion coefficients - a set of matrix equations. The absolute accuracy of molecular electronic structure calculations is ultimately determined by the quality of the basis set employed. No amount of configuration interaction will compensate for a poor choice of basis set. 

In the application of DHF to multipole polarizabilities, the first task is assembling the derivative Fock operators because in SCF the Fock operator is the effective (one-electron) Hamiltonian for the system. The parameters of differentiation are the elements of the expansion of the electrical potential, Vy, Fj, F, V y,.. ., . From Eqn. (28), one may see that the... 

In Volume 5 of this series, R. J. Bartlett and J. E Stanton authored a popular tutorial on applications of post-Hartree-Fock methods. Here in Chapter 2, Dr. T. Daniel Crawford and Professor Henry F. Schaefer III explore coupled cluster theory in great depth. Despite the depth, the treatment is brilliantly clear. Beginning with fundamental concepts of cluster expansion of the wavefunction, the authors provide the formal theory and the derivation of the coupled cluster equations. This is followed by thorough explanations of diagrammatic representations, the connection to many-bodied perturbation theory, and computer implementation of the method. Directions for future developments are laid out. 

In the next section we will discuss the approach we have developed for obtaining the molecular Hartree-Fock continuum orbitals. We will discuss how our approach is based on the Schwinger variational method and how in its present form it can be viewed as a hybrid method that uses both the basis-set expansion techniques of quantum chemistry and the numerical single-center expansion techniques of atomic collision physics. We will then discuss the results of applications of this approach to study shape resonances in the photolonlzatlon of several molecules, e.g., N2, CO, CO2, C2H2, and C2N2. These results will also be compared with available experimental data and with the results of studies of these same systems by different methods and models. 

A decade ago Laaksonen et al. published a paper giving an outline of the finite difference (FD) (or numerical) Hartree-Fock (HF) method for diatomic molecules and several examples of its application to a series of molecules (1). A summary of the FD HF calculations performed until 1987 can be found in (2). The work of Laaksonen et al. can be considered a second attempt to solve numerically the HF equations for diatomic molecules exactly. The earlier attempt was due to McCullough who in the mid 1970s tried to tackle the problem using the partial wave expansion method (3). This approach had been extended to study correlation effects, polarizabilities and hyper-fine constants and was extensively used by McCullough and his coworkers (4-6). Heinemann et al. (7-9) and Sundholm et al. (10,11) have shown that the finite element method could also be used to solve numerically the HF equations for diatomic molecules. 

In practical applications of the sapt approach to interactions of many-elect ron systems, one has to use the many-body version of sapt, which includes order-by-order the intramonomer correlation effects. The many-body SAPT is based on the partitioning of the total Hamiltonian as H = F+V+W, where the zeroth-order operator F = Fa + Fb is the sum of the Fock operators for the monomers A and B. The intermolecular interaction operator V = H — Ha — Hb is the difference between the Hamiltonians of interacting and noninteracting systems, and the intramonomer correlation operator W = Wa + Wb is the sum of the Moller-Plesset fluctuation potentials of the monomers Wx — Hx — Fx, X — A or B. The interaction operator V is taken in the non-expanded form, i.e., it is not approximated by the multipole expansion. The interaction energy components of Eq. (1) are now given in the form of a double perturbation series,... 

The Hartree-Fock method neglects the electron correlation. The reliability (and applicability) of the HF method itself is rather small. However, the electron correlation can be recovered by post-HF methods based on the wavefunction expansion in terms of Slater determinants constructed from the one-electron Hartree-Fock orbitals. 

If one works out this expression one obtains equations that are identical to equations (316) and (317). These equations were first derived by Talman and Shadwick [45]. Since in our procedure we optimized the energy of a Slater determinant wavefunction under the constraint that the orbitals in the Slater determinant come from a local potential, the method is also known as the optimized potential method (OPM). We have therefore obtained the result that the OPM and the expansion to order e2 are equivalent procedures. The OPM has many similarities to the Hartree-Fock approach. Within the Hartree-Fock approximation one minimizes the energy of a Slater determinant wavefunction under the constraint that the orbitals are orthonormal. One then obtains one-particle equations for the orbitals that contain a nonlocal potential. Within the OPM, on the other hand, one adds the additional requirement that the orbitals must satisfy single-particle equations with a local potential. Due to this constraint the OPM total energy ) will in general be higher than the Hartree-Fock energy Fhf, i.e., Ex > E. We refer to Refs. [46,47] for an application of the OPM method for molecules. 

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