Basis set, infinite

The curl condition given by Eq. (43) is in general not satisfied by the n x n matrix W (R ), if n does not span the full infinite basis set of adiabatic elechonic states and is huncated to include only a finite small number of these states. This tmncation is extremely convenient from a physical as well as computational point of view. In this case, since Eq. (42) does not have a solution, let us consider instead the equation obtained from it by replacing WC) (R t) by its longitudinal part... 

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. 

The parameterization of MNDO/AM1/PM3 is performed by adjusting the constants involved in the different methods so that the results of HF calculations fit experimental data as closely as possible. This is in a sense wrong. We know that the HF method cannot give the correct result, even in the limit of an infinite basis set and without approximations. The HF results lack electron correlation, as will be discussed in Chapter 4, but the experimental data of course include such effects. This may be viewed as an advantage, the electron correlation effects are implicitly taken into account in the parameterization, and we need not perform complicated calculations to improve deficiencies in fhe HF procedure. However, it becomes problematic when the HF wave function cannot describe the system even qualitatively correctly, as for example with biradicals and excited states. Additional flexibility can be introduced in the trial wave function by adding more Slater determinants, for example by means of a Cl procedure (see Chapter 4 for details). But electron cori elation is then taken into account twice, once in the parameterization at the HF level, and once explicitly by the Cl calculation. 

The calculated ioi as a function of basis set and electron correlation (valence electrons only) at the experimental geometry is given in Table 11.8. As the cc-pVXZ basis sets are fairly systematic in how they are extended from one level to the next, there is some justification for extrapolating the results to the infinite basis set limit (Section 5.4.5). The HF energy is expected to have an exponential behaviour, and a functional form of the type A + 5exp(—Cn) with n = 2-6 yields an infinite basis set limit of —76.0676 a.u., in perfect agreement with the estimated HF limit of -76.0676 0.0002 a.u. ... 

The model gives a unique answer in the limit of an infinite basis set, whereas density matrix fitting methods do not [10, 11]. 

The ideal calculation would use an infinite basis set and encompass complete incorporation of electron correlation (full configuration interaction). Since this is not feasible in practice, a number of compound methods have been introduced which attempt to approach this limit through additivity and/or extrapolation procedures. Such methods (e.g. G3 [14], CBS-Q [15] and Wl [16]) make it possible to approximate results with a more complete incorporation of electron correlation and a larger basis set than might be accessible from direct calculations. Table 6.1 presents the principal features of a selection of these methods. 

A third class of compound methods are the extrapolation-based procedures due to Martin [5], which attempt to approximate infinite-basis-set URCCSD(T) calculations. In the Wl method [16] calculations are performed at the URCCSD and URCCSD(T) levels of theory with basis sets of systematically increasing size. Separate extrapolations are then performed to determine the SCF, URCCSD valence-correlation, and triple-excitation components of the total atomization energy at... 

The second philosophy essentially views HF theoiy as a stepping stone on the way to exact solution of the Schrodinger equation. HF theory provides a very well defined energy, one which can be converged in the limit of an infinite basis set, and the difference between that... 

The basis set is the set of madiematical functions from which the wave function is constructed. As detailed in Chapter 4, each MO in HF theory is expressed as a linear combination of basis functions, the coefficients for which are determined from the iterative solution of the HF SCF equations (as flow-charted in Figure 4.3). The full HF wave function is expressed as a Slater determinant formed from the individual occupied MOs. In the abstract, the HF limit is achieved by use of an infinite basis set, which necessarily permits an optimal description of the electron probability density. In practice, however, one cannot make use of an infinite basis set. Thus, much work has gone into identifying mathematical functions that allow wave functions to approach the HF limit arbitrarily closely in as efficient a manner as possible. 

Solution of the HF equations with an infinite basis set is defined as the HF limit. Actually carrying out such a calculation is almost never a practical possibility. However, it is sometimes the case that one may extrapolate to the HF limit with a fair degree of confidence. 

Having discussed ways to reduce the scope of the MCSCF problem, it is appropriate to consider the other limiting case. What if we carry out a CASSCF calculation for all electrons including all orbitals in the complete active space Such a calculation is called full configuration interaction or full CF. Witliin the choice of basis set, it is the best possible calculation that can be done, because it considers the contribution of every possible CSF. Thus, a full CI with an infinite basis set is an exact solution of the (non-relativistic, Bom-Oppenheimer, time-independent) Schrodinger equation. 

Of course, there is a key difference between HF theory and DFT - as we have derived it so far, DFT contains no approximations it is exact. All we need to know is xc as a function of p. .. Alas, while Hohenberg and Kohn proved diat a functional of the density must exist. their proofs provide no guidance whatsoever as to its fonn. As a result, considerable research effort has gone into dnding functions of die density diat may be expected to reasonably approximate xc, and a discussion of diese is die subject of the next section. We close here by emphasizing that the key contrast between HF and DFT (in the limit of an infinite basis set) is that HF is a deliberately approximate theory, whose development was in part motivated by an ability to solve die relevant equations exactly, while DFT is an exact theory, but the relevant equations must be solved approximately because a key operator has unknown form. 

The magnetic field is independent of the choice of the gauge origin. So too are the computed magnetic properties if the wave function used is exact. Regrettably, we are not often afforded the opportunity to work with exact wave functions. For HF wave functions, one can also achieve independence of the gauge by using an infinite basis set, but that is hardly a practical option either. 

The set of initial atomic functions %i is called the basis set. Although the complete solution of the Hartree-Fock problem requires an infinite basis set, good approximations can be achieved with a limited number of atomic orbitals. The minimum number of such functions corresponding approximately to the number of electrons involved in the molecule is the minimal" basis set. The coefficients amt which measure the importance of each atomic orbital in the respective molecular orbitals are parameters determined by a variational procedure, i.e. chosen so as to minimize the expression... 

It also means that carrying out such calculations with a small basis set (double- or triple-, few or no polarization functions) can be a meaningless exercise, as one may be very far from the infinite basis set limit. Of course, in most cases, one uses calculations to determine the difference in energy between two species, for example, a... 

A measure of the extent to which any particular ab initio calculation does not deal perfectly with electron correlation is the correlation energy. In a canonical exposition [79] Lowdin defined correlation energy thus The correlation energy for a certain state with respect to a specified Hamiltonian is the difference between the exact eigenvalue of the Hamiltonian and its expectation value in the Hartree-Fock approximation for the state under consideration. This is usually taken to be the energy from a nonrelativistic but otherwise perfect quantum mechanical procedure, minus the energy calculated by the Hartree-Fock method with the same nonrelativistic Hamiltonian and a huge ( infinite ) basis set ... 

The general setting of the electronic structure description given above refers to a complete (and thus infinite) basis set of one-electron functions (spin-orbitals) (f>nwave functions, an additional assumption is made, which is that the orbitals entering eq. (1.136) are taken from a finite set of functions somehow related to the molecular problem under consideration. The most widespread approximation of that sort is to use the atomic orbitals (AO).17 This approximation states that with every problem of molecular electronic structure one can naturally relate a set of functions y/((r). // = M > N -atomic orbitals (AOs) centered at the nuclei forming the system. The orthogonality in general does not take place for these functions and the set y/ is characterized... 

This assumption however, is strictly only valid for infinite basis sets. 

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