Hartree-Fock virtual orbital

My personal special emphasis has always been on the wavefunction itself. Since the wavefunction is not an observable, it is not possible to carry out an empirical calibration of a model wavefunction. Rather one must place it in the context of a sequence of wavefunctions that ultimately converges to the exact answer and produces correct properties without empirical corrections. At the same time, I prefer wavefunctions that apply to as wide a range of molecular systems as possible but that have some chance of being interpreted. The Cl wavefunctions generated for small molecules using natural or MCSCF orbitals are of this type. More modern wavefunctions such as MPn, full Cl, or coupled clusters calculated with Hartree-Fock virtual orbitals are not interpretable, and are usually never even looked at. 

Exercise 4.8 For the special case of a two-electron system, the use of natural orbitals dramatically reduces the size of the full Cl expansion. If is the occupied Hartree-Fock spatial orbital and r = 2,3,..., K are virtual spatial orbitals, the normalized full Cl singlet wave function has the form... 

Hence, after a decade of false starts, chemists finally learned that the correct basis set should consist of functions that could represent the atomic Hartree-Fock orbitals plus allow for contraction and polarization corrections in the region where they are largest. Similarly it was realized that the Hartree-Fock virtual molecular orbitals were too diffuse for representing the correction to the SCF wavefunction due to electron correlation. Rather, correlation effects are best represented using excitations to nonphysical molecular orbitals that are of the same size as the occupied MOs. Initially this was learned by transforming existing wavefunctions to natural orbital form. Later, MCSCF orbital optimizations were used to obtain these localized correlating orbitals. 

Commutation occurs since the excitation operators always excite from the set of occupied Hartree-Fock spin orbitals to the virtual ones - see (13.1.2) for the double-excitation operators. The creation and annihilation operators of the excitation operators therefore anticommute. 

As before, the indices / and / are used for the occupied Hartree-Fock spin orbitals and the indices A and B for the unoccupied (virtual) spin orbitals. The cluster amplitudes tff are antisymmetric with respect to permutations of A and B and permutations of I and J. 

A Hbasis functions provides K molecular orbitals, but lUJiW of these will not be occupied by smy electrons they are the virtual spin orbitals. If u c were to add an electron to one of these virtual orbitals then this should provide a means of calculating the electron affinity of the system. Electron affinities predicted by Konpman s theorem are always positive when Hartree-Fock calculations are used, because fhe irtucil orbitals always have a positive energy. However, it is observed experimentally that many neutral molecules will accept an electron to form a stable anion and so have negative electron affinities. This can be understood if one realises that electron correlation uDiild be expected to add to the error due to the frozen orbital approximation, rather ihan to counteract it as for ionisation potentials. 

The il/j in Equation (3.21) will include single, double, etc. excitations obtained by promoting electrons into the virtual orbitals obtained from a Hartree-Fock calculation. The second-order energy is given by ... 

It should be noted that the Hartree-Fock equations F ( )i = 8i ([)] possess solutions for the spin-orbitals which appear in F (the so-called occupied spin-orbitals) as well as for orbitals which are not occupied in F (the so-called virtual spin-orbitals). In fact, the F operator is hermitian, so it possesses a complete set of orthonormal eigenfunctions only those which appear in F appear in the coulomb and exchange potentials of the Foek operator. The physical meaning of the occupied and virtual orbitals will be clarified later in this Chapter (Section VITA)... 

Configuration Interaction (Cl) methods begin by noting that the exact wavefunction 4 cannot be expressed as a single determinant, as Hartree-Fock theory assumes. Cl proceeds by constructing other determinants by replacing one or more occupied orbitals within the Hartree-Fock determinant with a virtual orbital. 

Here, occ means occupied and virt means virtual. In the restricted Hartree-Fock model, each orbital can be occupied by at most one a spin and one (i spin electron. That is the meaning of the (redundant) Alpha in the output. In the unrestricted Hartree-Fock model, the a spin electrons have a different spatial part to the spin electrons and the output consists of the HF-LCAO coefficients for both the a spin and the spin electrons. 

All three states were described by a single set of SCF molecular orbitals based on the occupied canonical orbitals of the X Z- state and a transformation of the canonical virtual space known as "K-orbitals" [10] which, among other properties, approximate the set of natural orbitals. Transition moments within orthogonal basis functions are easier to derive. For the X state the composition of the reference space was obtained by performing two Hartree-Fock single and double excitations (HFSD-CI) calculations at two typical intemuclear distances, i.e. R. (equilibrium geometry) and about 3Re,and adding to the HF... 

The difference between the Hartree-Fock energy and the exact solution of the Schrodinger equation (Figure 60), the so-called correlation energy, can be calculated approximately within the Hartree-Fock theory by the configuration interaction method (Cl) or by a perturbation theoretical approach (Mpller-Plesset perturbation calculation wth order, MPn). Within a Cl calculation the wave function is composed of a linear combination of different Slater determinants. Excited-state Slater determinants are then generated by exciting electrons from the filled SCF orbitals to the virtual ones ... 

At this point it should be noted that, in addition to the discussed previously, the canonical Hartree-Fock equations (26) have additional solutions with higher eigenvalues e . These are called virtual orbitals, because they are unoccupied in the 2iV-electron ground state SCF wavefunction 0. They are orthogonal to the iV-dimensional orbital space associated with this wavefunction. 

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