Wavefunction basis

The discussion presented in the subsequent parts of this chapter is based on the results of ab initio calculations of the electronic energy of molecular systems. Details about this kind of calculation are described in reference 11. In connection with this procedure, two major questions have to be addressed. The first is the choice of the wavefunction (basis set) to be used in the calculation, and the second whether or not to include electron correlation. 

Clementi, E., and Roetti, C. (1974). Roothaan-Hartree-Fock atomic wavefunctions basis functions and their coefficients for ground and certain excited states of neutral and ionized atoms, Z < 54. Atomic Data and Nuclear Data Tables 14, 177-320. 

In the multiple spawning method of Martfnez at al., the time-dependent Gaussian nuclear wavefunction basis x t) are supposed to be parametrized... 

If all the resonance states which fomi a microcanonical ensemble have random i, and are thus intrinsically unassignable, a situation arises which is caWtA. statistical state-specific behaviour [95]. Since the wavefunction coefficients of the i / are Gaussian random variables when projected onto (]). basis fiinctions for any zero-order representation [96], the distribution of the state-specific rate constants will be as statistical as possible. If these within the energy interval E E+ AE fomi a conthuious distribution, Levine [97] has argued that the probability of a particular k is given by the Porter-Thomas [98] distribution... 

A partial acknowledgment of the influence of higher discrete and continuum states, not included within the wavefunction expansion, is to add, to the tmncated set of basis states, functions of the fomi T p(r)<6p(r) where dip is not an eigenfiinction of the internal Flamiltonian but is chosen so as to represent some appropriate average of bound and continuum states. These pseudostates can provide fiill polarization distortion to die target by incident electrons and allows flux to be transferred from the the open channels included in the tmncated set. 

Once the requisite one- and two-electron integrals are available in the MO basis, the multiconfigurational wavefunction and energy calculation can begin. Each of these methods has its own approach to describing tlie configurations d),. j included m the calculation and how the C,.] amplitudes and the total energy E are to be... 

To remedy this diflSculty, several approaches have been developed. In some metliods, the phase of the wavefunction is specified after hopping [178]. In other approaches, one expands the nuclear wavefunction in temis of a limited number of basis-set fiinctions and works out the quantum dynamical probability for jumping. For example, the quantum dynamical basis fiinctions could be a set of Gaussian wavepackets which move forward in time [147]. This approach is very powerfLil for short and intemiediate time processes, where the number of required Gaussians is not too large. 

Ihe one-electron orbitals are commonly called basis functions and often correspond to he atomic orbitals. We will label the basis functions with the Greek letters n, v, A and a. n the case of Equation (2.144) there are K basis functions and we should therefore xpect to derive a total of K molecular orbitals (although not all of these will necessarily 3e occupied by electrons). The smallest number of basis functions for a molecular system vill be that which can just accommodate all the electrons in the molecule. More sophisti- ated calculations use more basis functions than a minimal set. At the Hartree-Fock limit he energy of the system can be reduced no further by the addition of any more basis unctions however, it may be possible to lower the energy below the Hartree-Fock limit ay using a functional form of the wavefunction that is more extensive than the single Slater determinant. 

In accordance with the variation theorem we require the set of coefficients that gives the lowest-energy wavefunction, and some scheme for changing the coefficients to derive that wavefunction. For a given basis set and a given functional form of the wavefunction (i.e. a Slater determinant) the best set of coefficients is that for which the energy is a minimum, at which point... 

Tie first consideration is that the total wavefunction and the molecular properties calculated rom it should be the same when a transformed basis set is used. We have already encoun-ered this requirement in our discussion of the transformation of the Roothaan-Hall quations to an orthogonal set. To reiterate suppose a molecular orbital is written as a inear combination of atomic orbitals ... 

Another approach is spin-coupled valence bond theory, which divides the electrons into two sets core electrons, which are described by doubly occupied orthogonal orbitals, and active electrons, which occupy singly occupied non-orthogonal orbitals. Both types of orbital are expressed in the usual way as a linear combination of basis functions. The overall wavefunction is completed by two spin fimctions one that describes the coupling of the spins of the core electrons and one that deals with the active electrons. The choice of spin function for these active electrons is a key component of the theory [Gerratt ef al. 1997]. One of the distinctive features of this theory is that a considerable amount of chemically significant electronic correlation is incorporated into the wavefunction, giving an accuracy comparable to CASSCF. An additional benefit is that the orbitals tend to be... 

As a final note, be aware that Hartree-Fock calculations performed with small basis sets are many times more prone to finding unstable SCF solutions than are larger calculations. Sometimes this is a result of spin contamination in other cases, the neglect of electron correlation is at the root. The same molecular system may or may not lead to an instability when it is modeled with a larger basis set or a more accurate method such as Density Functional Theory. Nevertheless, wavefunctions should still be checked for stability with the SCF=Stable option. ... 

A basis set is the mathematical description of the orbitals within a system (which in turn combine to approximate the total electronic wavefunction) used to perform the theoretical calculation. Larger basis sets more accurately approximate the orbitals by imposing fewer restrictions on the locations of the electrons in space. In the true quantum mechanical picture, electrons have a finite probability of existing anywhere in space this limit corresponds to the infinite basis set expansion in the chart we looked at previously. 

Run a second job with the 6-31G(d) basis set, starting from the STO-3G converged wavefunction. 

The next step on the road to quality is to expand the size of the atomic orbital basis set, and I hinted in Chapters 3 and 4 how we might go about this. To start with, we double the number of basis functions and then optimize their exponents by systematically repeating atomic HF-LCAO calculation. This takes account of the so-called inner and outer regions of the wavefunction, and Clementi puts it nicely. 

Simple Basis Set for Molecular Wavefunctions Containing First- and Second-Row Atoms E. Clementi... 

The first-order energy involves only the perturbation operator and the unperturbed wavefunction. In an HF-LCAO treatment, the integrals would be over the LCAOs, and this implies a four-index transformation to integrals over the basis functions. 

Notice that the Heilman-Feymnan theorem only applies to exact wavefunc-jB 8, not to variational approximations. All the enthusiasm of the 1960s and jWOs evaporated when it was realized that approximate wavefunctions them-Mves also depend on nuclear coordinates, since the basis functions are usually... 

In standard quantum-mechanical molecular structure calculations, we normally work with a set of nuclear-centred atomic orbitals Xi< Xi CTOs are a good choice for the if only because of the ease of integral evaluation. Procedures such as HF-LCAO then express the molecular electronic wavefunction in terms of these basis functions and at first sight the resulting HF-LCAO orbitals are delocalized over regions of molecules. It is often thought desirable to have a simple ab initio method that can correlate with chemical concepts such as bonds, lone pairs and inner shells. A theorem due to Fock (1930) enables one to transform the HF-LCAOs into localized orbitals that often have the desired spatial properties. 

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