Trial wave function

So far there have not been any restrictions on the MOs used to build the determinantal trial wave function. The Slater determinant has been written in terms of spinorbitals, eq. (3.20), being products of a spatial orbital times a spin function (a or /3). If there are no restrictions on the form of the spatial orbitals, the trial function is an Unrestricted Hartree-Fock (UHF) wave function. The term Different Orbitals for Different Spins (DODS) is also sometimes used. If the interest is in systems with an even number of electrons and a singlet type of wave function (a closed shell system), the restriction that each spatial orbital should have two electrons, one with a and one with /3 spin, is normally made. Such wave functions are known as Restricted Hartree-Fock (RHF). Open-shell systems may also be described by restricted type wave functions, where the spatial part of the doubly occupied orbitals is forced to be the same this is known as Restricted Open-shell Hartree-Fock (ROHF). For open-shell species a UHF treatment leads to well-defined orbital energies, which may be interpreted as ionization potentials. Section 3.4. For an ROHF wave function it is not possible to chose a unitary transformation which makes the matrix of Lagrange multipliers in eq. (3.40) diagonal, and orbital energies from an ROHF wave function are consequently not uniquely defined, and cannot be equated to ionization potentials by a Koopman type argument. 

The parameterization of MNDO/AM1/PM3 is performed by adjusting the constants involved in the different methods so that the results of HF calculations fit experimental data as closely as possible. This is in a sense wrong. We know that the HF method cannot give the correct result, even in the limit of an infinite basis set and without approximations. The HF results lack electron correlation, as will be discussed in Chapter 4, but the experimental data of course include such effects. This may be viewed as an advantage, the electron correlation effects are implicitly taken into account in the parameterization, and we need not perform complicated calculations to improve deficiencies in fhe HF procedure. However, it becomes problematic when the HF wave function cannot describe the system even qualitatively correctly, as for example with biradicals and excited states. Additional flexibility can be introduced in the trial wave function by adding more Slater determinants, for example by means of a Cl procedure (see Chapter 4 for details). But electron cori elation is then taken into account twice, once in the parameterization at the HF level, and once explicitly by the Cl calculation. 

The HF method determines the best one-determinant trial wave function (within the given basis set). It is therefore clear that in order to improve on HF results, the starting point must be a trial wave function which contains more than one Slater Determinant (SD). This also means that the mental picture of electrons residing in orbitals has to be abandoned, and the more fundamental property, the electron density, should be considered. As the HF solution usually gives 99% of the correct answer, electron correlation methods normally use the HF wave function as a starting point for improvements. 

A generic multi-determinant trial wave function can be written as... 

This is perhaps the easiest method to understand. It is based on the variational principle (Appendix B), analogous to the HF method. The trial wave function is written as a linear combination of determinants with the expansion coefficients determined by requiring that the energy should be a minimum (or at least stationary), a procedure known as Configuration Interaction (Cl). The MOs used for building the excited Slater determinants are taken from a Hartree-Fock calculation and held fixed. Subscripts S, D, T etc. indicate determinants which are singly, doubly, triply etc. excited relative to the... 

In other words, the exact wave function behaves asymptotically as a constant 4- l/2ri2 when ri2 is small. It would therefore seem natural that the interelectronic distance would be a necessary variable for describing electron correlation. For two-electron systems, extremely accurate wave functions may be generated by taking a trial wave function consisting of an orbital product times an expansion in electron coordinates such as... 

The variation principle then says that the energy E0 of the ground state is the lower bound of the quantity Eq. II.6 for arbitrary normalized trial wave functions W and that further all eigenfunctions satisfy the relation... 

So the first iteration transforms the trial wave functions expressed as linear combinations of gaussian functions into an expression which involves Dawson functions [62,63], We have not been able to find a tabular entry to perform explicitly the normalization of the first iterate, accordingly this is carried out numerically by the Gauss-Legendre method [64],... 

Therefore P and P respectively, are different, and we can use VP as trial wave function for H. We must then have by virtue of the variational principle... 

Stated in still other words this means that for any trial density p(r) - which satisfies the necessary boundary conditions such as p( ) - 0, J p( ) dr = N, and which is associated with some external potential Vext - the energy obtained from the functional given in equation (4-6) represents an upper bound to the true ground state energy E0. E0 results if and only if the exact ground state density is inserted into equation (4-8). The proof of the inequality (4-11) is simple since it makes use of the variational principle established for wave functions as detailed in Chapter 1. We recall that any trial density p(r) defines its own Hamiltonian H and hence its own wave function. This wave function can now be taken as the trial wave function for the Hamiltonian generated from the true external potential Vext. Thus, we arrive at... 

In practice, using trial wave functions with one or more variable parameters, the true ground state may be approached by minimizing the expectation value of the energy with respect to these parameters. 

HF method determines the energetically best determinantal trial wave function (4>o) and this would be improved further by including more configurations. Let [Pg.30]

The trial wave functions of a Schrodinger equation are expressed as determinant of the HF orbitals. This will give coupled nonlinear equations. The amplitudes were solved usually by some iteration techniques so the cc energy is computed as... 

Thus, in spite of choosing the form (9) for a trial wave function and the form (10) for a partition of the Hamiltonian we do not recover an expression... 

Iosio Kato in 1957. [92] Unfortunately, any trial wave function composed of Slater determinants has smooth first and higher derivatives with respect to the interelec-tronic coordinates. Thus, even though such expansions are insightful and preserve the concept of orbitals to some extent, from a mathematical point of view they are expected to be slowly convergent. 

Equation (1.10) represents the Hartree Hamiltonian and Eq. (1.8) has to be solved by iteration, in the sense that a guessed trial wave function 1) is introduced in Eq. (1.10) and the Schrodinger equation Eq. (1.8) solved. The resulting wave function is again introduced in Eq. (1.10) and Eq. (1.8) is again solved until self-consistency is achieved. 

Of course the cusp can be represented by including the interparticle distances in a trial wave functions, most simply by means of Jastrow factors, exponentials of the inter-particle distances. But the problems of integral evaluation with such fac-... 

To get some idea of the use of trial wave functions and the variation principle, evaluate the expectation value of the energy using the Hydrogen atom Hamiltonian, and normalized Is orbitals with variable Z. That is, evaluate ... 

The overlap of ip with the true ground state eigenfunction ipo is greater than or equal to 1 — e that is, the spatial distribution of the trial wave function is a very good approximation to the true wave function, and... 

The approach used first, historically, and the one this book is about, is called the valence bond (VB) method today. Heitler and London[8], in their treatment of the H2 molecule, used a trial wave function that was appropriate for two H atoms at long distances and proceeded to use it for all distances. The ideal here is called the separated atom limit . The results were qualitatively correct, but did not give a particularly accurate value for the dissociation energy of the H—H bond. After the initial work, others made adjustments and corrections that improved the accuracy. This is discussed folly in Chapter 2. A cmcial characteristic of the VB method is that the orbitals of different atoms must be considered as nonorthogonal. 

The density matrices are by definition partial integrals of the corresponding trial wave functions FcQrs ( ti Xi, X2,..., xjv) obtained for the given composition C and nuclear configuration Q so that they have the specified total spin S and spatial symmetry T ... 

In ab initio methods the HER approximation is used for build-up of initial estimate for and which have to be further improved by methods of configurational interaction in the complete active space (CAS) [39], or by Mpller-Plesset perturbation theory (MPn) of order n, or by the coupled clusters [40,41] methods. In fact, any reasonable result within the ab initio QC requires at least minimal involvement of electron correlation. All the technical tricks invented to go beyond the HFR calculation scheme in terms of different forms of the trial wave function or various perturbative procedures represent in fact attempts to estimate somehow the second term of Eq. (5) - the cumulant % of the two-particle density matrix. 

The computational problem, then, is determination of the cluster amplitudes t for aU of the operators included in tlie particular approximation. In the standard implementation, this task follows the usual procedure of left-multiplying the Schrodinger equation by trial wave functions expressed as dctcnninants of the HF orbitals. This generates a set of coupled, nonlinear equations in the amplitudes which must be solved, usually by some iterative technique. With the amplitudes in hand, the coupled-cluster energy is computed as... 

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