Fixed basis functions

You should remember the basic physical idea behind the HF model each electron experiences an average potential due to the other electrons (and of course the nuclei), so that the HF Hamiltonian operator contains within itself the averaged electron density due to the other electrons. In the LCAO version, we seek to expand the HF orbitals i/ in terms of a set of fixed basis functions X X2 > and write... 

This equation corresponds to a unitary transformation from the space-fixed basis functions jf(R,r) to the body-fixed functions J,K,M,p)QfK Q)-The inverse transforamtion is... 

This is the key equation for calculating the HF electronic energy of a molecule. It can be used when self-consistency has been reached, or after each SCF cycle employing the e s and c s yielded by that particular iteration, and E , which latter does not change from iteration to iteration, since it is composed only of the fixed basis functions and an operator which does not contain e s or c s from Eqs. 5.64 = 5.19 and 5.79... 

The electronic Coulomb interaction u(r 12) = greatly complicates the task of formulating and carrying out accurate computations of iV-electron wave functions and their physical properties. Variational methods using fixed basis functions can only with great difficulty include functions expressed in relative coordinates. Unless such functions are present in a variational basis, there is an irreconcilable conflict with Coulomb cusp conditions at the singular points ri2 - 0 [23, 196], No finite sum of product functions or Slater determinants can satisfy these conditions. Thus no practical restricted Hilbert space of variational trial functions has the correct structure of the true V-electron Hilbert space. The consequence is that the full effect of electronic interaction cannot be represented in simplified calculations. 

Table 1-8. Quantum numbers and space-fixed basis functions for different dimers. The entries a , d and p in columns 1 and 2 refer to atom, diatomic, and polyatomic molecule, respectively...

In interpolating methods it is possible to differentiate between fixed basis functions (i.e. linear, cubic or thin-plate splins) and basis functions with adjustable parameters (kriging). Furthermore, kriging has a statistical interpretation that allows the construction of estimations or the error in the interpolator, which can be crucial in the development of an accurate optimization algorithm. Due to these adjustable parameters kriging interpolation tends to produce the better results[2,3]... 

Before presenting the linear method, let us briefly review how the energy-band problem has been tackled in the past. In this context we note that the traditional methods may be divided into those which express the wave functions as linear combinations of some fixed basis functions, say plane waves or atomic orbitals, and those like the cellular, APW, and KKR methods which employ matching of partial waves. As we shall see, both approaches have their strong and weak points. 

The linear methods devised by Andersen [1.19] are characterised by using fixed basis functions constructed from partial waves and their first energy derivatives obtained within the muffin-tin approximation to the potential. 

The representation of trial functions as linear combinations of fixed basis functions is perhaps the most common approach used in variational calculations optimization of the coefficients is often said to be an application of the linear variational principle. Although some very accurate work on small atoms (notably helium and lithium) has been based on complicated trial functions with several nonlinear parameters, attempts to extend these calculations to larger atoms and molecules quickly runs into formidable difficulties (not the least of which is how to choose the form of the trial function). Basis set expansions like that given by equation (A1.1.113) are much simpler to design, and the procedures required to obtain the coefficients that minimize s are all easily carried out by computers. 

The primitive body-fixed basis functions as described above do not have definite parity, except for K = 0. However, since parity is known to be a rigorously good quantum number, it is usually advantageous to choose basis functions which do have definite parity, and it is straightforward to define linear combinations of the primitive functions for which this is the case. Adopting the notation 0 = K, these are... 

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. ... 

This approximation has the immense advantage of reducing the number of integrals to be calculated, and we could in principle calculate the remainder of them exactly if we knew which basis functions were involved. When Pariser and Parr first tried to calculate the excitation energies of unsaturated hydrocarbons on the assumption that the basis functions Xi were ordinary orbitals, they got very poor agreement with experiment. But when they treated the integrals as parameters that had to be fixed by appeal to experiment, they got much better agreement. 

For bonded atoms, the off-diagonal terms (where i j) are taken to depend on tjje type and length of the bond joining the atoms on which the basis functions y- and Xj 0 centred. The entire integral is written as a constant, 0ij, which is not the same as the fixY in Hiickel 7r-electron theory. The are taken to be parameters, fixed by calibration against experiment. It is usual to set Pij to zero when the pair of atoms are not formally bonded. 

Thus a hydrogen atom is represented by two basis functions, the first of which is a fixed linear combination of three primitives, and the other one a more diffuse primitive as shown in Table 9.6. 

Consider now making the variational coefficients in front of the inner basis functions constant, i.e. they are no longer parameters to be determined by the variational principle. The Is-orbital is thus described by a fixed linear combination of say six basis functions. Similarly the remaining four basis functions may be contracted into only two functions, for example by fixing the coefficient in front of the inner three functions. In doing this the number of basis functions to be handled by the variational procedure has been reduced from 10 to three. 

Combining the full set of basis functions, known as the primitive GTOs (PGTOs), into a smaller set of functions by forming fixed linear combinations is known as basis set contraction, and the resulting functions are called contracted GTOs (CGTOs). 

Finally we observe from Fig. 1 the magnitude of Gw qz) decreases for increasing at fixed w. Thus, the only way to fulfill the Bethe sum rule at arbitrarily large values of q will be to include basis functions of arbitrarily high angular momentum. This confirms a previously reached conclusion [12],... 

Fixed-shape basis functions. The basis functions are of a fixed shape, such as linear, sigmoid, Gaussian, wavelet, or sinusoid. Adjusting the... 

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