Gaussian linear combination

In the collapse phase the monomer density p = N/R is constant (for large N). Thus, the only confonnation dependent tenn in (C2.5.A1) comes from the random two-body tenn. Because this tenn is a linear combination of Gaussian variables we expect that its distribution is also Gaussian and, hence, can be specified by the two moments. Let us calculate the correlation i,) / between the energies and E2 of two confonnations rj ]and ry jof the chain in the collapsed state. The mean square of E is... 

Here, the force F is a linear combination of the components of R it also has a Gaussian distribution and autocorrelation matrix that satisfies the same properties of R t) as shown in eq. (3), with I (the nxn unit matrix) replacing M [71] ... 

The solution to this problem is to use more than one basis function of each type some of them compact and others diffuse, Linear combinations of basis Functions of the same type can then produce MOs with spatial extents between the limits set by the most compact and the most diffuse basis functions. Such basis sets arc known as double is the usual symbol for the exponent of the basis function, which determines its spatial extent) if all orbitals arc split into two components, or split ualence if only the valence orbitals arc split. A typical early split valence basis set was known as 6-31G 124], This nomenclature means that the core (non-valence) orbitals are represented by six Gaussian functions and the valence AOs by two sets of three (compact) and one (more diffuse) Gaussian functions. 

The second approximation in HF calculations is due to the fact that the wave function must be described by some mathematical function, which is known exactly for only a few one-electron systems. The functions used most often are linear combinations of Gaussian-type orbitals exp(—nr ), abbreviated GTO. The wave function is formed from linear combinations of atomic orbitals or, stated more correctly, from linear combinations of basis functions. Because of this approximation, most HF calculations give a computed energy greater than the Hartree-Fock limit. The exact set of basis functions used is often specified by an abbreviation, such as STO—3G or 6—311++g. Basis sets are discussed further in Chapters 10 and 28. 

Chapter 10 represented a wave function as a linear combination of Gaussian basis functions. Today, there are so many basis sets available that many researchers will never need to modify a basis set. However, there are occasionally times when it is desirable to extend an existing basis set in order to obtain more accurate results. The savvy researcher also needs to be able to understand the older literature, in which basis sets were customized routinely. 

The contracted Gaussian functions are a linear combination of the primitive Gaussian functions. That is,... 

T vo main streams of computational techniques branch out fiom this point. These are referred to as ab initio and semiempirical calculations. In both ab initio and semiempirical treatments, mathematical formulations of the wave functions which describe hydrogen-like orbitals are used. Examples of wave functions that are commonly used are Slater-type orbitals (abbreviated STO) and Gaussian-type orbitals (GTO). There are additional variations which are designated by additions to the abbreviations. Both ab initio and semiempirical calculations treat the linear combination of orbitals by iterative computations that establish a self-consistent electrical field (SCF) and minimize the energy of the system. The minimum-energy combination is taken to describe the molecule. 

We have extended the linear combination of Gaussian-type orbitals local-density functional approach to calculate the total energies and electronic structures of helical chain polymers[35]. This method was originally developed for molecular systems[36-40], and extended to two-dimensionally periodic sys-tems[41,42] and chain polymers[34j. The one-electron wavefunctions here are constructed from a linear combination of Bloch functions c>>, which are in turn constructed from a linear combination of nuclear-centered Gaussian-type orbitals Xylr) (in ihis case, products of Gaussians and the real solid spherical harmonics). The one-electron density matrix is given by... 

Her workers to fit the exchange-correlation potential and the charge density (in the Coulomb potential) to a linear combination of Gaussian-typc 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 ... 

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 ... 

The next step was to represent each Slater atomic orbital as a fixed linear combination of Gaussian orbitals so a Slater-type orbital with exponent f is written as a sum of GTOs with exponents a, q 2, and so on. For example, in the case of three primitive GTOs we might write... 

An extended basis set of atomic functions expressed as fixed linear combinations of Gaussian functions is presented for hydrogen and the... 

If a 33 = 0, we have a i3 = 0, and the function 03 is then a linear combination of the functions 0X and 0 2 and should be omitted in the orthogonalization process, which is here simply accomplished by means of the Gaussian elimination technique developed for solving equation systems. The connection between the matrices a and a may be written in the form ... 

The method presented here allows, starting with trial gaussian functions, a partial analytical treatment which we have used to improve the LCAO-GTO orbitals (trial functions) essentially obtained from all ab initio quantum chemistry programs. As in r-representation, trial functions (t>i( Hp) (Eq. 21) are conveniently expressed as linear combinations of m functions Xi(P) themselves written as linear combinations of Gt gaussian functions (LCAO-GTO approximation) gta(P). 

The various quantities entering Eq. 24 are deduced when the trial orbitals ( )i( >(p) are expressed as linear combinations of Gaussian functions, they are expressible in terms of... 

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],... 

For atoms, the radial part of ( )8i(p) is expressed as a linear combination of spherical gaussians, which, in the case of 2p orbitals writes as ... 

Haberlen, O.D. and Rdsch, N. (1992) A scalar-relativistic extension of the linear combination of Gaussian-type orbitals local density functional method application to AuFl, AuCl and Au2. Chemical Physics Letters, 199, 491-496. 

---