Primitive basis function Gaussian

To incorporate the angular dependence of a basis function into Gaussian orbitals, either spherical haimonics or integer powers of the Cartesian coordinates have to be included. We shall discuss the latter case, in which a primitive basis function takes the form... 

Next, we shall consider four kinds of integrals. The first is the expectation value of the Coulomb potential by one nucleus for one of the primitive basis function centered at that nucleus. The second is the expectation value of the Coulomb potential by one nucleus for one of the primitive basis function centered at a different point (usually another nucleus). Then, we will consider the matrix element of a Coulomb term between two primitive basis functions at different centers. The third case is when one basis function is centered at the nucleus considered. The fourth case is when both basis functions are not centered at that nucleus. By that we mean, for two Gaussian basis functions defined in Eqs. (73) and (74), we are calculating... 

Figure 4-7 The SCF energy of the neon atom converges exponentially with the number of Gaussian primitive basis functions.

E Matrix elements of the truncated harmonic potential F Matrix elements in Gaussian primitive basis functions... 

F Matrix elements in Gaussian primitive basis functions... 

An expansion of the Morse potential, for example, in a set of Gaussian functions is given by eq (C4) in Appendix C. Matrix elements of the Morse potential in terms of the Gaussian primitive basis functions are therefore simply three center overlap integrals [49], These matrix elements can be evaluated for each term in the sum and then converted to the final expression in a straightforward manner. 

A typical basis function is a hxed linear combination of simpler, primitive functions. Such a composite function is termed a contracted basis function. Each primitive basis function is centered at an atomic nucleus and has a Gaussian dependence on distance from that nucleus. Except for s-functions, it also has a Cartesian factor to describe its angular dependence. Eor example, a p primitive function looks like xexp(—(r ), where ( (zeta) is the exponent of the Gaussian function. A p contracted function is a hxed linear combination of two or more primitive functions with different exponents. 

Illogical as it might seem, most modem day solutions of the molecular electronic Schrbdinger equation use, as the primitive basis functions in the calculation, a set of functions which do not satisfy the boundary conditions expected for the solution of the equation. They do not behave properly in the region very near the nucleus, nor do they behave properly in regions far from the nucleus. These primitive functions, the set of Gaussian functions defined by... 

The functions put into the determinant do not need to be individual GTO functions, called Gaussian primitives. They can be a weighted sum of basis functions on the same atom or different atoms. Sums of functions on the same atom are often used to make the calculation run faster, as discussed in Chapter 10. Sums of basis functions on different atoms are used to give the orbital a particular symmetry. For example, a water molecule with symmetry will have orbitals that transform as A, A2, B, B2, which are the irreducible representations of the C2t point group. The resulting orbitals that use functions from multiple atoms are called molecular orbitals. This is done to make the calculation run much faster. Any overlap integral over orbitals of different symmetry does not need to be computed because it is zero by symmetry. 

Minimal basis sets use fixed-size atomic-type orbitals. The STO-3G basis set is a minimal basis set (although it is not the smallest possible basis set). It uses three gaussian primitives per basis function, which accounts for the 3G in its name. STO stands for Slater-type orbitals, and the STO-3G basis set approximates Slater orbitals with gaussian functions. ... 

The actual basis functions are formed as linear combinations of such primitive gaussians ... 

The columns to the right of the first vertical line of asterisks hold the exponents (a above) and the coefficients (the d p s) for each primitive gaussian. For example, basis function 1, an s function, is a linear combination of six primitives, constructed with the exponents and coefficients (the latter are in the column labeled S-COEF ) listed in the table. Basis function 2 is another s function, comprised of three primitives using the exponents and S-COEF coefficients from the section of the table corresponding to functions 2-5. Basis function 3 is a p function also made up or three primitives constructed from the exponents and P-COEF coefficients in the same section of the table ... 

The coefficients specified for the component primitive gaussians are chosen so that the resulting constructed basis functions are normalized. This means that one coefficient in each set is effectively constrained so that this condition is fulfilled. 

Linear combinations of primitive gaussians like these are used to form the actual basis functions the latter are called contracted gaussians and have the form ... 

In general, all 17 s-primitives contribute to each s-derived molecular orbital. Obviously, the tighter Gaussians will contribute more strongly to the inner-shell molecular orbitals and the more diffuse Gaussians to the valence i-orbitals. Nevertheless, it is impossible (and also not desired) to make a connection between basis functions and atomic orbitals. 

---