Expansion of the Wavefunction

Consider a model system of four electrons moving in an arbitrary electrostatic field generated by the nuclei in a molecule. For our purposes, it is not necessary to specify the number of these nuclei, their types, or positions only the general form of the electronic wavefunction is of interest. It is convenient to describe the motions of each electron separately by assigning them to one-electron functions, (l),(xi), where Xi is a vector of the coordinates (including spin) of electron 1. In addition, electrons are fermions, so the electronic wave-function must be antisymmetric with respect to interchange of the coordinates of any pair of electrons. A traditional and very useful starting point for such a four-electron wavefunction is the so-called Slater determinant 

The component functions t may be chosen in a variety of ways. For example, if the nuclear field were only a single beryllium nucleus, the one-electron spatial functions could be constructed to mimic the atomic Is and 2s orbitals. For a molecular system, the functions can be constructed as a linear combination of atomic orbitals (AOs) in which each one-electron function 

How can we improve this so-called independent-particle approximation such that the motions of the electrons are correlated Often the set of occupied orbitals (i.e., those functions that compose the Slater determinant above) is chosen from a larger set of one-electron functions. These extra functions are frequently referred to as virtual orbitals and may, for example, arise as a byproduct of the SCF procedure. Within the space described by the full set of orbitals, any function of N variables may be written in terms of N-tuple products of the (j)p. For example, a function of two variables may be constructed by using all possible binary products of the set of one-electron functions  

In addition, the determinantal form of the individual terms in this expansion implies antisymmetrization of the cluster coefficients, such that Iff- = -tf-j = 

It should be carefully noted here that the cluster function, / y(xi, X2), is intended to correlate the motions of any pair of electrons placed in orbitals i and and not just the motions of electrons 1 and 2. Since the Slater determinant produces a linear combination of orbital products, including terms such as 

A molecular multipole expansion is poorly converged at distances of chemical interest, for instance, at distances between atoms of molecules in a dimer or crystal. To obtain electrostatic interaction energy between heteronuclear diatomic molecules at close distance it is always better to model observed molecular point dipoles by a distributed dipole model that places net charges on the atoms. In general, a distributed multipole model of a molecule consists of a set of sites with each site having its own multipoles. Obviously distributed multipole models are not uniquely defined. For instance, at which site in a polyatomic ion is the ionic charge (monopole) located Do you spread it around evenly among the sites  

The electric potential resulting from any charge distribution can be represented by a convergent multicenter multipole expansion.Molecular moments of isolated molecules can be obtained experimentally, such as from the Stark effect on the microwave spectra of gas molecules. Studies of van der Waals clusters and molecular association in general require a multicenter model for short-range electrostatic interaction. Mulder and Huiszoon, for instance, found that a molecular multipole expansion was not satisfactory for the electrostatic interaction of 

The multipole moments of the charge distribution described by the ab initio wavefunction are defined as the expectation value of the corresponding oper- 

Thus any molecular multipole moment may be decomposed into additive atomic multipole moments (AAMMs) 

The hydrogen fluoride molecule may be used as a simple example of this approach. Using the 6-31G basis the PA charge on F is -0.395 e. This monopole value yields, when combined with the bond distance of 1.733 au, a dipole moment of 0.685. However, the exact operator dipole obtained from the wavefunction is 0.776. In the cumulative procedure, the difference between these two values is made up with atomic dipoles. From [13] one obtains (w)f = + 0.330 and [w] = +0.446, which of course sum to the exact molecular dipole. Application of Eq. [14] yields cumulative atomic dipoles Mj. = -0.012 and Mh = 0.103. The sum of the PA charge dipole plus the atomic dipoles equals the exact molecular dipole. Analogous procedures are used for higher moments. 

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To this end we make a perturbation expansion of the wavefunctions and energies... 

The theory behind body-fixed representations and the associated angular momentum function expansions of the wavefunction (or wave packet) in terms of bases parameterized by the relevant constants of the motion and approximate constants of the motion is highly technical. Some pertinent results will simply be stated. The two good constants of the motion are total angular momentum, J, and parity, p = +1 or 1. An approximate constant of the motion is K, the body-fixed projection of total angular momentum on the body-fixed axis. For simplicity, we will restrict attention to the helicity-decoupled or centrifugal sudden (CS) approximation in which K can be assumed to be a constant of the motion. In terms of aU its components, and the iteration number k, the real wave packet is taken to be [21]... 

Nakatsuji H, Hirao K (1978) Cluster expansion of the wavefunction. symmetry-adapted-cluster expansion, its variational determination, and extension of open-shell orbital theory. J Chem Phys 68 2053... 

In order to relate the dressed state population dynamics to the more intuitive semiclassical picture of a laser-driven charge oscillation, we analyze the induced dipole moment n) t) and the interaction energy V)(0 of the dipole in the external field. To this end, we insert the solution of the TDSE (6.27) into the expansion of the wavefunction Eq. (6.24) and determine the time evolution of the charge density distribution p r, t) = -e r, f)P in space. Erom the density we calculate the expectation value of the dipole operator... 

Configuration interaction has come to mean any expansion of the wavefunction in a finite series of N-electron functions (28)... 

Bianco et al. [23] proposed a direct VB wavefunction method combined with a PCM approach to study chemical reactions in solution. Their approach is based on a Cl expansion of the wavefunction in terms of VB resonance structures, treated as diabatic electronic states. Each diabatic component is assumed to be unchanged by the interaction with the solvent the solvent effects are exclusively reflected by the variation of the coefficients of the VB expansion. The advantage of this choice is related to its easy interpretability. The method has been applied to the study of the several SN1/2 reactions. 

Nakatsuji, H. (1983). Cluster expansion of the wavefunction. Valence and Rydberg excitations, ionization and inner-valence ionizations of COj and NjO studied by the SAC and SAC Cl theories. Chem. Phys. 75, 425-41. 

The pseudopotential concept was advanced a long ago [1] and is based on the natural energetic and spatial separation of core and valence electrons. The concept allows a significant reduction in computational efforts without missing the essential physics of phenomena provided the interaction of core and valence electrons is well described by some effective (model) Hamiltonian. Traditionally, pseudopotentials are widely used in the band structure calculations [2], because they allow convenient expansions of the wavefunctions in terms of plane waves suited to describing periodical systems. For molecular and/or nonperiodical systems, the main advantage of pseudopotentials is a... 

H. Nakatsuji and K. Hirao, J. Chem. Phys., 68, 2053 (1978). Cluster Expansion of the Wavefunction. Symmetry-Adapted-Cluster Expansion, Its Variational Determination, and Extension of Open-Shell Orbital Theory. 

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. 

The APW (augmented plane wave) method was devised by Slater (1937,1965), and is based on the solution of the Schrodinger equation for a spherical periodic potential using an expansion of the wavefunction in terms of solutions of the atomic problem near the nucleus, and an expansion in plane waves outside a predetermined sphere in the crystal. 

Depending on the relative sizes of the number of electrons, the number of orbitals, and the excitation level, one can derive several different simple estimates of the computational cost of a configuration interaction procedure. Obviously that cost relates to the number of Ar-electron functions in the linear expansion of the wavefunction, and the size of the Cl space for various methods has already been discussed in section 2.4.1. 

The set of AO basis functions is collected into the row vector / and a column of the matrix C is the set of MO expansion coefficients for a particular molecular orbital matrix operations results from the expansion of the wavefunction 0> in a set of A-electron expansion functions... 

The first energy expression of the above type results from truncating the expansion of the wavefunction variations to include only the first-order changes... 

In 1966, Silverstone and Sinanoglu [77] and Kelly [80] published their extension of previous formal developments and applications of fhe cluster expansion of the wavefunction and MBPT for closed-(sub)shell ground states to analogous formalisms for open-(sub)shell sfafes. In facf, Kelly demonstrated his methods with an impressive calculation of parfs of elec-fron correlation in the oxygen atom, with emphasis on the correlation of pairs of electrons [80]. Among other things, he pointed to the inevitable appearance of terms that correspond to spin-orbital pair excitations such as 2p(-Fl ),2p(0+ 2p(-l+),4/(-F2+, which are called semi-internal by Silverstone and Sinanoglu [77], see below. 

The Coupled-clusters (CC) method[7] based on the cluster expansion of the wavefunction has been established as a highly reliable method for calculations of ground state properties of small molecules with the spectroscopic accuracy. When this method is used together with a flexible basis set it recovers the dominant part of the electron correlation. Typically, CC variant explicitly considering single and double excitations (CCSD) is used. In order to save computer time the contributions from triple excitations are often calculated at the perturbation theory level (notation CCSD(T) is used in this case). CCSD(T) method can be routinely used only for systems with about 10 atoms at present. Therefore, it cannot be used directly in zeolite modeling, however, results obtained at CCSD(T) level for small model systems can serve as an important benchmark when discussing the reliability of more approximate methods. 

The values for these parameters and the ratios between them can be compared to those given by Shabaev (Eqs. (13)-(15) in [35]) from a series expansion of the wavefunctions for a homogeneously charged nucleus with radius R The ratio between the parameters for ks and are then given by 62/02 = 3/5, 64/04 = 3/7 and 65/06 = 3/9 (in reasonable agreement with the results from the fit). The importance of the higher correction from the series expansion is found by Shabaev [35] to be... 

In most standard ab initio approaches, the parameters to minimize are the linear coefficients of the expansion of the wavefunction in some basis set. To make the problem tractable, one is usually forced to choose a basis set for which the integrals of Eq. (4.1) are analytically computable. However, as we have seen, it is practical to use very accurate explicitly correlated wavefunctions with VMC. 

However, the expansion of the wavefunction in terms of billions of determinants is of little conceptual or interprctational value. There are two possible views on this latter fact ... 

As we shall see, the most common use of the variation method is not to find a set of linear parameters in the determinantal expansion of the wavefunction but to model the electronic structure and optimise the parameters contained in the mathematical formulation of that model. 

In this article we will take the opportunity to derive the expansion for the wavefunction in a heuristic, and we hope accessible, fashion. We refer the reader to references [8] and [9] for a more thorough treatment of the expansion of the wavefunction and further details about the interdimensional degeneracies. 

The fact that one method may include more terms in an expansion them another method does not necessarily imply that it is superior. The terms which are left out of an expansion of the wavefunction or expectation value are, in fact, often just as important as the ones which are actually included [111,119],... 

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