Unoccupied orbital

An additional important factor affecting the bonding of the heavier group 13 elements is the limited number of valence electrons available for bond formation. In neutral molecules the use of the three valence electrons to form three electron pair bonds necessarily leaves a valence orbital unoccupied. This usually results in association or, in cases where one of the bonds involves another group 13 metal, a disproportionation reaction such as that shown in Eq. (1). 

Figure 16.1 The chemical hardness of an atom, molecule, or ion is defined to be half. The value of the energy gap between the bonding orbitals (HOMO—highest orbitals occupied by electrons), and the anti-bonding orbitals (LUMO—lowest orbitals unoccupied by electrons). The zero level is the vacumn level, so I is the ionization energy, and A is the electron affinity, (a) For hard molecules the gap is large (b) it is small for soft molecules. The solid circles represent valence electrons. Adapted from Atkins (1991).

The particle-hole formalism has been introduced as a simplihcation of many-body perturbation theory for closed-shell states, for which a single Slater determinant dominates and is hence privileged. One uses the labels i,j, k,... for spin orbitals occupied in <1> and a,b,c,... for spin orbitals unoccupied virtual) in . 

In order to discuss the valence electronic state and chemical bonding of lithium vanadium oxide we made calculations using cluster models. The density of states (DOS) and the partial density of states (PDOS) of Li1.1V0.9O2 are obtained by this study as shown in Figure 3.4. The filled band located from —8 to —3 eV is mainly composed of O 2p orbital. The partially filled band located around —2 to 4 eV is mainly composed of V 3d orbital. Unoccupied band located above 5 eV is... 

To improve upon a double- basis, one generally adds polarization functions whose / values correspond to orbitals unoccupied in the free atoms. For example, to expand upon a double- basis for HjO, one would add 2p functions on H and 3d functions on O. Such functions are called polarization functions since they describe the polarization of atomic electron density arising from molecule formation. For example, if a H atom is placed in the electrostatic field of an O atom, its electron density will be polarized along the 0-H bond direction, a change that can be described by the mixing of H 2p character into the His wave function. 

Diatomic molecules are still rather simple systems suitable for clarifying problems encountered in polyatomic molecules. Moreover, they give us additional information not obtainable from atomic calculations, Polarization functions represent a typical example. Since they correspond to orbitals unoccupied in atomic ground states, their exppnents cannot be estimated from atomic calculations. Among the calculations on diatomic molecules, very useful information on the problem of basis set composition was contributed by calculations reported by Cade, Huo, Wahl, Sales, Liu, Yoshimine and others " . Consider for example the results for the nitrogen molecules. Fig. 2.2 presents the dependence of the total energy and its components (ki-... 

Downward-directed lines represent hole states (orbitals occupied in the reference) and upward directed lines represent particle states (orbitals unoccupied in the reference). Hence, one may interpret Figure 1(d) as a single-determinant wavefunction that differs from the reference by a single excitation from orbital (j), to orbital Furthermore, this convention implies that the reference wave-function itself is represented by empty space, as indicated in Figure 1(c). 

In contrast to Rydberg orbitals, unoccupied (virtual) valence orbitals (V) are relatively compact, being made up of valence-shell principal-quantum-number atomic orbitals. These arise from the splittings of valence-shell atomic orbitals that occur upon bond formation. As a consequence of their compact natures, excitations between occupied and unoccupied valence orbitals can have large transition moments. Since each molecule has only a small number of such orbitals, it is a simple matter to determine the V class of molecular orbitals for any compound from the atomic shells of the atoms comprising the molecule of Interest, and from the total number of electrons that fill the molecular shells ( ). 

In this simple scheme, such other eigenstates of Hq as are formed by replacing one of the molecular orbitals ground state, call it ipjj), that is also an eigenfunction of S f/j. Thus we would have... 

Because the matrix P or F has dimension M x M, we obtain M a-spin and M P-spin spatial orbitals via diagonalization. But we only use and of these orbitals to construct the SCF solution. This leaves us with M - = N and M - Kp = Np spin orbitals unoccupied in the SCF solution. We identify these orbitals collectively by the letters a, b,c,, whereas i, j, k,. . . indicate orbitals occupied in the SCF solution, and p, q,r,s,... indicate any orbital. The orbitals a, b, c,. . . are frequently called virtual orbitals because they... 

Following the customary terminology, we will call inactive holes the inactive occupied orbitals, doubly filled in every model CSF. The inactive particles will refer to aU the orbitals unoccupied in every CSF. Orbitals which are occupied in some (singly or doubly) but unoccupied in others are the active orbitals. In our spin-free form, the labels are for orbitals only, and not for spin orbitals. From the mode of definition, no active orbital can be doubly occupied in every model CSF. We want to express the cluster operator T, inducing excitations to the virtual functions, in terms of excitations of minimum excitation rank, and at the same time wish to represent them in a manifestly spin-free form. To accomplish this, we take as the vacuum—for excitations out of 4> — the largest closed-shell portion of it, For each such vacuum, we redefine the holes and particles, respectively, as ones which are doubly occupied and unoccupied in < 0 a-The holes are denoted by the labels. .., etc. and the particle orbitals are denoted as a, etc. The particle orbitals are totally unoccupied in any or are necessarily... 

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