Hartree-Fock basis set

The hydrogen fluoride riddle was solved and the era of molecular Hartree-Fock computations was open. The need to have adequately large, polarized and balanced basis sets, is tantamount to require near to Hartree-Fock basis sets. The work was quickly written, submitted and printed [33] it was the first Hartree-Fock type computation for a many electron molecule with the clear message not to waste computer time and human effort with minimal basis sets. Well,... this was a rather strong message Indeed, minimal basis sets computations littered (embellished ) the computational chemistry literature for decades to follow, with the more or less acceptable excuse that computer time can be too expensive or not available. 

Hartree-Fock basis set "Transition State" basis set ... 

Model Parameters -> Computational implementation Results Reality Hartree-Fock Basis set Various cutoffs Total energies o Atomization energy... 

The approach discussed here has its origins in both many-body theory and multichannel scattering theory. From many-body theory it builds on the method called "configuration interaction" or "superposition of configurations" (SOC), especially as that method is viewed as a systematic way to enlarge multiconfiguration Hartree-Fock basis sets until convergence is reached. This... 

Table 8.12 The Hartree-Fock eneigy hf and the CISD correlation energy faso — hf for the ground state of the carbon atom calculated using correlation-consistent basis sets. All energies are given in atomic units. The Hartree-Fock basis-set limit is —37.688619 Eh and the estimated CISD correlation-energy basis-set limit is —99.7 0.4 mEh...

Table 8.15 The Hartree-Fock energy and the CISD valence correlation energy for the ground state of the carbon atom and for the 5 ground state of the carbon anion calculated using the cc-pVXZ and aug-cc-pVXZ basis sets (atomic units). The Hartree-Fock basis-set limits for the neutral and anionic atoms are —37.688619 and —37.708844 Eh, respectively. The estimated CISD basis-set limits are —99.7 0.4 mEh and —120.7 0.4 mEh, respectively...

The Dirac-Hartree-Fock iterative process can be interpreted as a method of seeking cancellations of certain one- and two-body diagrams.33,124 The self-consistent field procedure can be regarded as a sequence of rotations of the trial orbital basis into the final Dirac-Hartree-Fock orbital set, each set in this sequence forming a basis for the Furry bound-state interaction picture of quantum electrodynamics. The self-consistent field potential involves contributions from the negative energy states of the unscreened spectrum so that the Dirac-Hartree-Fock method defines a stationary point in the space of possible configurations, rather that a variational minimum, as is the case in non-relativistic theory. 

Up to this point we have focused on determinants and configurations formed from a set of canonical Hartree-Fock orbitals. The resulting Cl expansion unfortunately turns out to be rather slowly convergent. It is clear, however, that one can perform a Cl calculation using N-electron configurations formed from any one-electron basis. Therefore, it is of interest to ask whether one can find a one-electron basis for which the Cl expansion is more rapidly convergent than it is with the Hartree-Fock basis, and thus be able to obtain equivalent results with a smaller number of configurations. The set of natural orbitals, introduced by P.-O. Lowdin, forms such a basis. 

Although in principle it is possible to work with the optimized state in the representation of the original orbitals (10.2.2), it is much more convenient to express the Hartree-Fock state directly in terms of a set of MOs where HF) corresponds to ic = 0. This is easily achieved by transforming the elementary operators (the MOs) to the Hartree-Fock basis... 

Ihe one-electron orbitals are commonly called basis functions and often correspond to he atomic orbitals. We will label the basis functions with the Greek letters n, v, A and a. n the case of Equation (2.144) there are K basis functions and we should therefore xpect to derive a total of K molecular orbitals (although not all of these will necessarily 3e occupied by electrons). The smallest number of basis functions for a molecular system vill be that which can just accommodate all the electrons in the molecule. More sophisti- ated calculations use more basis functions than a minimal set. At the Hartree-Fock limit he energy of the system can be reduced no further by the addition of any more basis unctions however, it may be possible to lower the energy below the Hartree-Fock limit ay using a functional form of the wavefunction that is more extensive than the single Slater determinant. 

The application of density functional theory to isolated, organic molecules is still in relative infancy compared with the use of Hartree-Fock methods. There continues to be a steady stream of publications designed to assess the performance of the various approaches to DFT. As we have discussed there is a plethora of ways in which density functional theory can be implemented with different functional forms for the basis set (Gaussians, Slater type orbitals, or numerical), different expressions for the exchange and correlation contributions within the local density approximation, different expressions for the gradient corrections and different ways to solve the Kohn-Sham equations to achieve self-consistency. This contrasts with the situation for Hartree-Fock calculations, wlrich mostly use one of a series of tried and tested Gaussian basis sets and where there is a substantial body of literature to help choose the most appropriate method for incorporating post-Hartree-Fock methods, should that be desired. 

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. 

It is a well-known fact that the Hartree-Fock model does not describe bond dissociation correctly. For example, the H2 molecule will dissociate to an H+ and an atom rather than two H atoms as the bond length is increased. Other methods will dissociate to the correct products however, the difference in energy between the molecule and its dissociated parts will not be correct. There are several different reasons for these problems size-consistency, size-extensivity, wave function construction, and basis set superposition error. 

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