Closed-shell ground states

To define the state yon want to calculate, you must specify the m u Itiplicity. A system with an even ii n m ber of electron s n sn ally has a closed-shell ground state with a multiplicity of I (a singlet). Asystem with an odd niim her of electrons (free radical) nsnally has a multiplicity of 2 (a doublet). The first excited state of a system with an even ii nm ber of electron s usually has a m n Itiplicity of 3 (a triplet). The states of a given m iiltiplicity have a spectrum of states —the lowest state of the given multiplicity, the next lowest state of the given multiplicity, and so on. 

The chemical reactivity of radicals is governed of course by the same chemical principles as the reactivity of systems having closed-shell ground states. Both equilibrium and rate processes are important here. The paucity of quantitative data on equilibrium and rate constants of radical reactions, suitable from the viewpoint of the present state of the theory, prevents a more rapid development in the MO applications this difficulty, however, is not specific for open-shell systems. 

The most straightforward way of using quantum-chemical information at the atomistic level is by means of a force field. This requires that no truly electronic processes (chemical reactions, electronically excited states) are involved and that basically a description of a (macro)molecule in its closed-shell ground state is desired. 

Two isolated reactant molecules in the closed-shell ground state are designated as A and B, whose electronic energies are IFao and Wbo, respectively. Here the term closed-shell implies the structure of a molecule with doubly occupied MO s only. The lowest total energy of the two mutually interacting systems is denoted by W. Then, the interaction energy is defined by... 

The ionization energy Ij of a closed-shell ground-state molecule M is defined as the energy needed to yield M+ in its electronic state 2 Py according to... 

The set of trial determinants wiU of course include the closed-shell ground-state determinant ... 

For the closed shell ground state, Sp includes ( )o. Furthermore, for the same state, but in the MR-SDCI cases, other c ) are included in Sp, but their weight in the wave function will always remain lower than unity. Hence, we can write, in the intermediate normalization... 

The second approach to calculating MCD starts from its definition in terms of the real part of first-order correction to the frequency-dependent polarizability in the presence of a magnetic field (Section II.A.6). This definition can be used to consider all types of MCD linear in the magnetic field (9). Our current implementation is restricted to systems with a closed-shell ground state. We shall therefore only consider the calculation of A and terms by this method. 

Figure 2.41. The closed-shell ground-state leads to a repulsive interaction with the substrate. An internal bond is broken to form the bond to the substrate. This bond-prepared adsorbate state corresponds to an excited gas-phase triplet state. From Ref. [84].

CASSCF wave functions based on an active space formed by two orbitals (a central Mg-2p orbital and an orbital with strong central Mg-3s character) and by two electrons are constructed for a singlet closed-shell ground state and singlet and triplet excited states. The CASSCF result is improved by perturbation theory... 

Each Prs involves the sum over the occupied MO s (j = 1 -n we are dealing with a closed-shell ground-state molecule with 2n electrons) of the products of the coefficients of the basis functions 4>r and cf)s. As pointed out in Section 5.2.3.6.2 the Hartree-Fock procedure is usually started with an initial guess at the coefficients. We can use as our guess the extended Hiickel coefficients we obtained for HeH+, with this same geometry (Section 4.4.1.2) we need the c s only for the occupied MO s ... 

The method of calculating wavefunctions and energies that has been described in this chapter applies to closed-shell, ground-state molecules. The Slater determinant we started with (Eq. 5.12) applies to molecules in which the electrons are fed pairwise into the MO s, starting with the lowest-energy MO this is in contrast to free radicals, which have one or more unpaired electrons, or to electronically excited molecules, in which an electron has been promoted to a higher-level MO (e.g. Fig. 5.9, neutral triplet). The Hartree-Fock method outlined here is based on closed-shell Slater determinants and is called the restricted Hartree-Fock method or RHF method restricted means that the electrons of a spin are forced to occupy (restricted to) the same spatial orbitals as those of jl spin inspection of Eq. 5.12 shows that we do not have a set of a spatial orbitals and a set of [l spatial orbitals. If unqualified, a Hartree-Fock (i.e. an SCF) calculation means an RHF calculation. 

The essence of the hole-particle formalism lies in a new meaning given the vacuum state. Consider now the closed shell ground state Slater determinante I d>0 > expressed by means of Eq. (25) as... 

We use the second quantization formalism to express the second order contribution to the correlation energy of the closed-shell ground state. By substituting the expression (67) for W in Eq. (72) we obtain... 

Here we shall demonstrate how to obtain the explicit expression129 13°1 for (188) by means of the MB-RSPT. Let us study the ionized state which we shall describe using I ki>. We shall limit ourselves to a state I for which the initial state li) is obtained from the neutral closed shell ground state in which we annihilate one particle. The state l< j) is therefore realized by... 

In this chapter we shall present the fundamental ideas of the MB-RSPT in its application to the interaction of two closed-shell ground state molecular systems. Remarkable features of the approach are the following ... 

The excited state of a diamagnetic species with a closed-shell ground state is both a better donor and acceptor than its associated ground state. In addition to the ground-state redox potential, the excited state has the additional redox power of the absorbed photon, i.e. hv. Figure 2.15 illustrates this point, whereby ionization potentials (IPs) and electron affinities (EAs) for a ground- and excited-state molecule are compared. Excitation decreases the IP by AEhomo-lumo/ i.e. 

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