Improving the Atomic Orbital

The horizontal axis corresponds to an intemuclear separation running from 1.5 to 2.5 ao and the vertical axis corresponds to an orbital exponent running from 1.0 to 1.4. The potential energy minimum corresponds to an exponent of 1.238, and we note the contraction of the atomic Is orbital on molecule formation. 

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Each of these tools has advantages and limitations. Ab initio methods involve intensive computation and therefore tend to be limited, for practical reasons of computer time, to smaller atoms, molecules, radicals, and ions. Their CPU time needs usually vary with basis set size (M) as at least M correlated methods require time proportional to at least M because they involve transformation of the atomic-orbital-based two-electron integrals to the molecular orbital basis. As computers continue to advance in power and memory size, and as theoretical methods and algorithms continue to improve, ab initio techniques will be applied to larger and more complex species. When dealing with systems in which qualitatively new electronic environments and/or new bonding types arise, or excited electronic states that are unusual, ab initio methods are essential. Semi-empirical or empirical methods would be of little use on systems whose electronic properties have not been included in the data base used to construct the parameters of such models. 

To improve our model we note that s- and /7-orbitals are waves of electron density centered on the nucleus of an atom. We imagine that the four orbitals interfere with one another and produce new patterns where they intersect, like waves in water. Where the wavefunctions are all positive or all negative, the amplitudes are increased by this interference where the wavefunctions have opposite signs, the overall amplitude is reduced and might even be canceled completely. As a result, the interference between the atomic orbitals results in new patterns. These new patterns are called hybrid orbitals. Each of the four hybrid orbitals, designated bn, is formed from a linear combinations of the four atomic orbitals ... 

The energy difference is expected to be reduced and perhaps eliminated when the calculations are repeated utilizing the 4-3IG basis set because of Pople s experience with the C3H7 ion. The non-classical form should be preferentially stabilized by the improved representation of the atomic orbitals, which should enable calculations to give a better account of the long bonds and three-centre ring. 

The solution of the secular equation Fy —F5y = 0 requires the evaluation of the constituent matrix terms Fy. The Fy s are, however, themselves functions of the coefficients of the atomic orbitals amt through Pjel and therefore can only be evaluated by solving the secular equation. The Hartree-Fock procedure thus requires that a preliminary guess be made as to the values of the molecular population distribution terms Pici these values are then used to calculate the matrix elements Fy and thence solve the secular determinant. This, in turn, provides a better approximation to the wave function and an. .improved set of values of Pm. The above procedure is repeated with this first improved set and a second improved set evaluated. The process is repeated until no difference is found between successive improved wave functions. Finally, it may be shown that when such a calculation has been iterated to self-consistency the total electronic energy E of a closed shell molecule is given by... 

Technically, the simultaneous optimization of orbitals and coefficients for a multistructure VB wave function can be done with the VBSCF method due to Balint-Kurti and van Lenthe [21,22], The VBSCF method has the same format as the classical VB method with an important difference. While the classical VB method uses orbitals that are optimized for the separate atoms, the VBSCF method uses a variational optimization of the atomic orbitals in the molecular wave function. In this manner the atomic orbitals adapt themselves to the molecular environment with a resulting significant improvement in the total energy and other computed properties. 

Solve the Fock matrix eigenvalue equations given above to obtain the orbital energies and an improved occupied molecular orbital. In so doing, note that the normalization condition <0i i> = 1 = ]SCi gives the needed normalization condition for the expansion coefTicients of the 0i in the atomic orbital basis. 

Consider some improvements on the Heitler-London function (13.101). One obvious step is the introduction of an orbital exponent C in the Is function. This was done by Wang in 1928. The optimum value of C is 1.166 at Rg, and and R are improved to 3.78 eV and 0.744 A. Recall that Dickinson in 1933 improved the Finkelstein-Horowitz HJ trial function by mixing in some 2p character into the atomic orbitals (hybridization). In 1931 Rosen used this idea to improve the Heitler-London-Wang function. He took the trial function... 

On the other hand, the linear combination of atomic orbitals - molecular orbital (LCAO-MO) theory, is actually the same as Hartree-Fock theory. The basic idea of this theory is that a molecular orbital is made of a linear combination of atom-centered basis functions describing the atomic orbitals. The Hartree-Fock procedure simply determines the linear expansion coefficients of the linear combination. The variables in the Hartree-Fock equations are recursively defined, that is, they depend on themselves, so the equations are solved by an iterative procedure. In typical cases, the Hartree-Fock solutions can be obtained in roughly 10 iterations. For tricky cases, convergence may be improved by changing the form of the initial guess. Since the equations are solved self-consistently, Hartree-Fock is an example of a self-consistent field (SCF) method. 

An efficient approach to improve on the Hartree-Fock Slater determinant is to employ Moller-Plesset perturbation theory, which works satisfactorily well for all molecules in which the Dirac-Hartree-Fock model provides a good approximation (i.e., in typical closed-shell single-determinantal cases). The four-component Moller-Plesset perturbation theory has been implemented by various groups [519,584,595]. A major bottleneck for these calculations is the fact that the molecular spinor optimization in the SCF procedure is carried out in the atomic-orbital basis set, while the perturbation expressions are given in terms of molecular spinors. Hence, all two-electron integrals required for the second-order Moller-Plesset energy expression must be calculated from the integrals over atomic-orbital basis functions like... 

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