Expansion of the Wave Function

In this section we shall discuss an approach which is neither variational nor perturbational. This approach has its origin in nuclear physics and was introduced to quantum chemistry by Sinanoglu, It is based on a cluster expansion of the wave function. A systematic method for calculation of cluster expansion components of the exact wave function was developed by CiSek. The characteristic feature 

We shall express the exact wave function in the form 

In equations (4.24) the t operators are generating the clusters U from the Slater determinant P and their effect on may be viewed as follows  

Let us compare this cluster expansion with the well known Cl expansion of the wave function 

We see that the cluster expansion is formally the same, only instead of the c-set of expansion coefficients we have the d-coefficients (appearing in eqns. (4.25)-(4.27))4 Comparing coefficients standing before respective configurations gives us the following relations  

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When the HF wave function gives a very poor description of the system, i.e. when nondynamical electron correlation is important, the multiconfigurational SCF (MCSCF) method is used. This method is based on a Cl expansion of the wave function in which both the coefficients of the Cl and those of the molecular orbitals are variationally determined. The most common approach is the Complete Active Space SCF (CASSCF) scheme, where the user selects the chemically important molecular orbitals (active space), within which a full Cl is done. 

Considering the different calculated values for an individual complex in Table 11, it seems appropriate to comment on the accuracy achievable within the Hartree-Fock approximation, with respect to both the limitations inherent in the theory itself and also to the expense one is willing to invest into basis sets. Clearly the Hartree-Fock-Roothaan expectation values have a uniquely defined meaning only as long as a complete set of basis functions is used. In practice, however, one is forced to truncate the expansion of the wave function at a point demanded by the computing facilities available. Some sources of error introduced thereby, namely ghost effects and the inaccurate description of ligand properties, have already been discussed in Chapter II. Here we concentrate on the... 

As shown in Table VII, our best results for LiH and LiD electron affinities obtained with the 3600-term expansions of the wave functions for LiH /LiH and LiD /LiD are 0.33030 and 0.32713 eV, respectively. Even though, as stated, both values represent lower bounds to the tme EAs, they both are within the uncertainty brackets of the experimental results of 0.342 0.012 eV (LiH) and 0.337 0.012 eV (LiD) obtained by Bowen and co-workers [126]. 

In SOPPA [5] a Mpller-Plesset perturbation theory expansion of the wave function [28,48] is employed ... 

The harmonic potential in the second term in Eq. (3.88) is necessary to avoid unwanted compression or expansion of the wave function in the x-y plane. However, the driving potential in Eq. (3.88) does not satisfy the Laplace equation. The difficulty is overcome by combining this electromagnetic field and the field that compresses (expands) a wave function in a harmonic potential in the z-direction in a fashion that suppresses unwanted excitation. Below we examine the rotation... 

For the general case with Nq states in Q, it follows directly that the coefficients for the states in the expansion of the wave function c,E ) in H in eigenstates of Q and V are given by Shapiro [34]... 

In this section we shall discuss an approach which is neither variational nor perturba-tional. This approach also has its origin in nuclear physics and was introduced to quantum chemistry by Sinanoglu47, It is based on a cluster expansion of the wave function. A systematic method for the calculation of cluster expansion components of the exact wave function was developed by C ek48 The characteristic feature of this approach is the expansion of the wave function as a linear combination of Slater determinants. Formally, this expansion is similar to the ordinary Cl expansion. The cluster expansion, however, gives us not only the physical insight of the correlation energy but it also shows the connections between the variational approaches (Cl) and the perturbational approaches (e.g. MB-RSPT). 

In Section V we presented a method which is neither perturbational nor variational but which possesses certain features of both. Here, we shall make a few remarks on another promising method of this kind. Instead of the cluster expansion of the wave function this method is based on the idea of direct combination of the perturbation expansion with the variational method. 

The general setting of the electronic structure description given above refers to a complete (and thus infinite) basis set of one-electron functions (spin-orbitals) (f>nwave functions, an additional assumption is made, which is that the orbitals entering eq. (1.136) are taken from a finite set of functions somehow related to the molecular problem under consideration. The most widespread approximation of that sort is to use the atomic orbitals (AO).17 This approximation states that with every problem of molecular electronic structure one can naturally relate a set of functions y/((r). // = M > N -atomic orbitals (AOs) centered at the nuclei forming the system. The orthogonality in general does not take place for these functions and the set y/ is characterized... 

Second-order perturbations are characterized by the necessary expansion of the wave function, and within simplifying fictitious MO models interaction between symmetry-equivalent orbitals will cause a split into an antibonding and a bonding linear combination. This orbital mixing increases both with the square of the interaction parameter, 0, and with decreasing energy difference Aa. For radical cation states of a molecule, in which the... 

Fig. 5. The graphical representation of the direct renormalization approach. The triangle with the letter n inside means the expansion of the wave function for the bound electron state n in terms of free electron wave functions...

A general approach to the intramonomer correlation problem is known as the many-electron (or many-body) SAPT method88,141 213-215. In this method the zeroth-order Hamiltonian H0 is decomposed as H0 = F + W, where F = FA + FB is the sum of the Fock operators, FA and FB, of monomer A and B, respectively, and W is the intramonomer correlation operator. The correlation operator can be written as W = WA + WB, where Wx = Hx — Fx, X = A or B. The total Hamiltonian can be now be represented as H = F + V + W. This partitioning of H defines a double perturbation expansion of the wave function and interaction energy. In the SRS theory the wave function is obtained by expanding the parametrized Schrodinger equation as a power series in and A,... 

The 1inked-cluster theorem for energy, from the above analysis, is a consequence of the connectivity of T, and the exponential structure for ft. Size-extensivity is thus seen as a consequence of cluster expansion of the wave function. Specfic realizations of the situation are provided by the Bruckner—Goldstone MBPT/25,26/, as indicated by Hubbard/27/, or in the non-perturbative CC theory as indicated by Coester/30,31/, Kummel/317, Cizek/32/, Paldus/33/, Bartlett/21(a)/ and others/30-38/. There are also the earlier approximate many-electron theories like CEPA/47/, Sinanoglu s Many Electron Theory/28/ or the Cl methods with cluster correction /467. 

Notice that p A = A p, since V A = 0. The terms in A are to be treated as a perturbation, but we immediately see a difficulty in that A is proportional to r and becomes very large in a large system. That difficulty was removed for a tight-binding expansion of the wave function by White (1974) we use a similar but more direct approach here instead of writing the electron states in the crystal as a... 

We may estimate the ion softening in perturbation theory as in Section 14-B, using the expansion of the wave function to first order in the interatomic matrix elements, Eq. (14-4), which we rewrite... 

Sinanoglu was the first who suggested a practical method for calculating the correlation energy based on the cluster expansion of the wave function. By the approximate treatment of the problem (4,42) he arrived for the function... 

This is a multi-center expansion of the wave function in terms of the incoming waves of the system. It can also be expressed as a single center expansion... 

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