Open-shell states

The open-shell states are of ungerade symmetry and all the minimal-basis CSFs have different space and spin symmetries see Section 5.2.2. The open-shell CSFs therefore represent our final approximate states for the model system. The energies of these states are obtained from the expressions 

We now proceed to consider the magnetic interactions involving the electron spin S in E states with open shell electronic structures. The magnetic dipole moment arising from electron spin is 

With the introduction of electronic angular momeutum, we have to consider how the spin might be coupled to the rotational motion of the molecule. This question becomes even more important when electronic orbital angular momentum is involved. The various coupling schemes give rise to what are known as Hund s coupling cases they are discussed in detail in chapter 6, and many practical examples will be encountered elsewhere in this book. If only electron spin is involved, the important question is whether it is quantised in a space-fixed axis system, or molecule-fixed. In this section we confine ourselves to space quantisation, which corresponds to Himd s case (b). 

We deal first with molecules containing one unpaired elechon (,S = 1/2) where magnetic nuclei are not present. The electron spin magnetic moment then interacts with the magnetic moment due to molecular rotation, the interaction being represented by the Hamiltonian term 

The lower rotational levels for a case (b) state are shown in figure 1.7(b). The spin-rotation interaction takes the same form as for a state, given in 

In appendix 8.3 we show that (1.48) with q = 0 leads to the simple expression. 

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For reasons given later, we shall most frequently use in applications the method of Longuet-Higgins and Pople. Recently, the half-electron method was extended to the lowest-energy open-shell states of any given symmetry and multiplicity (57). 

Obviously, vanishes, unless Up + Uq + rir + tig = 2. Sufficient though not necessary for Eq. (176) is that the Up are equal to 0,, or 1 that is, now open-shell states with fractional NSO occupation numbers are also possible. This implies via Eq. (174) that the only nonvanishing elements of >-2 are those... 

For OH and SH, the NOF EAs are larger than the experimental values. This trend is due to the expected underestimation of the correlation energy for open-shell states with our approach. In fact, we fix the unpaired electron in the corresponding HF higher-occupied molecular orbital (HOMO) of the neutral molecule, and then this level does not participate in the correlation. Note that for these molecules the total spin of the neutral molecule is greater than the total spin of the anion (S > Sa)- The underestimation of the total energy is for neutral molecules larger than for anions and therefore the NOF vertical EAs are overestimated. 

K the atomic-like state hybridizes with band states, its eigenvalue broadens to a well defined bandwidth W. The electron becomes more itinerant (the UAV ratio being a good parameter to describe this trend - see Chap. A), and the screening becomes poorer. Moreover, the relative intensities of the two peaks in the split core level responses depend on the occupation probability of the pulled down states, which is enhanced when hybridization becomes larger. In fact, attempts have been made to establish correlations between the intensity ratio of the two peaks of the split response and locaUzation or hybridization of the open shell states in a transition series. 

For quantum chemistry, first-row transition metal complexes are perhaps the most difficult systems to treat. First, complex open-shell states and spin couplings are much more difficult to deal with than closed-shell main group compounds. Second, the Hartree—Fock method, which underlies all accurate treatments in wavefunction-based theories, is a very poor starting point and is plagued by multiple instabilities that all represent different chemical resonance structures. On the other hand, density functional theory (DFT) often provides reasonably good structures and energies at an affordable computational cost. Properties, in particular magnetic properties, derived from DFT are often of somewhat more limited accuracy but are still useful for the interpretation of experimental data. 

We now consider the theoretical calculation of excited-state wave functions. This is more difficult than ground-state calculations because we are dealing with open-shell configurations. The Hartree-Fock equations for a state of an open-shell configuration have a more complicated form than for closed shells, and there exist close to a dozen different approaches to excited-state Hartree-Fock calculations. As noted earlier, the Hartree-Fock wave function for a closed-shell state is a single determinant, but for open-shell states, we may have to take a linear combination of a few Slater determinants to get a Hartree-Fock function that is an eigenfunction of S and Sz and has the correct spatial symmetry. 

As a first application of a new analytical gradient method employing UHF reference functions, seven different methods for inclusion of correlation effects were employed to optimize the geometry and calculate the harmonic vibrational frequencies and dipole moments of the lowest open-shell states for three simple hydrides including 3Z i SiH2228. As the degree of correlation correction increased, results approached those from the best multiconfiguration SCF calculation. 

The possibility of N2 coordination to up to four (six) iron atoms has been proposed by Dance on the basis of restricted frozen-core Kohn-Sham calculations on a FeMoco model (32,33). It was found that a binding mode intermediate between p Vn and p4,r 2 coordination to be most stable. However, these propositions are not necessarily the final answer since open-shell states are most likely to become important and exact exchange was not present in the density functional chosen to cure the singlet preference of the pure density functional (cf. discussion in the Appendix). 

For applications to open-shell states of atoms and molecules or to metallic solids, and for systems at finite temperature, it is convenient to treat occupation numbers as parameters that can vary freely in the range 0 < n, < 1. If the occupation numbers... 

The quadrupole coupling is very much the most important nuclear hyperfine interaction in1 + states, and it takes the same form in open shell states as in closed shells. We turn now to the much smaller interactions involving magnetic dipole moments, two types of which may be present. A nuclear spin I gives rise to a magnetic moment fih... 

Open shell states with both spin and orbital angular momentum... 

The form of the complete Zeeman effective Hamiltonian for a diatomic molecule in a given vibrational level of an open shell state has been given by Brown, Kaise, Kerr and Milton [30], It is the sum of the following terms ... 

SEMI-CLUSTER EXPANSION THEORIES FOR THE OPEN-SHELL STATES... 

There is an intimate connection between the cluster-expansion of a wave-function and the property of size-extensivity. To describe this aspect in the simplest manner,it is pertinent to recall first the closed-shell ground state. The ways to encompass the open-shell states can then be indicated as appropriate extensions and generalizations of the closed-shell cluster expansion strategy. 

As explained in Sec.2, the full cluster—expansion theories in Hilbert space are designed to compute wave-functions for the open-shell states that are explicitly size-extensive with respect to the total number of electrons N. The underlying cluster structure of all these developments is what was envisaged by Silverstone and Sinanoglu/48/s... 

Thus, the main relativistic effects are (1) the radical contraction and energetic stabilization of the s and p orbitals which in turn induce the radial expansion and energetic destabilization of the outer d and f orbitals, and (2) the well-known spin-orbit splitting. These effects will be pronounced upon going from As to Sb to Bi. Associated with effect (1), it is interesting to note that the Bi atom has a tendency to form compounds in which Bi is trivalent with the 6s 6p valence configuration. For this tendency of the 6s electron pair to remain formally unoxidized in bismuth compounds (i.e. core-like nature of the 6s electrons), the term inert pair effect or nonhybridization effect has been often used for a reasonable explanation. In this context, the relatively inert 4s pair of the As atom (compared with the 5s pair of Sb) may be ascribed to the stabilization due to the d-block contraction , rather than effect (1) . On the other hand, effect (2) plays an important role in the electronic and spectroscopic properties of atoms and molecules especially in the open-shell states. It not only splits the electronic states but also mixes the states which would not mix in the absence of spin-orbit interaction. As an example, it was calculated that even the ground state ( 2 " ) of Bij is 25% contaminated by Hg. In the Pauli Hamiltonian approximation there is one more relativistic effect called the Dawin term. This will tend to counteract partially the mass-velocity effect. 

X. Li and J. Paldus, ]. Chem. Phys., 102, 2013 (1995). Spin-Adapted Open-Shell State-Selective Coupled-Cluster Approach and Doublet Stability of Its Hartree-Fock Reference. 

It is not necessary to base the calculation on a singlet closed-shell reference state, but the vast majority of chemical applications choose this option. Open-shell reference EOM-CC and EOM-CC methods are, however, relevant to the discussion of real singularities, since it is the (ROHF)UHF-EOM-CC excitation energies relative to the open-shell state of interest that give the energy differences in the denominators of Eq. (18). 

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