Electron states singly occupied

Friedrich Hund determined a set of rules to determine the ground state of a multi-electron atom in the 1920s. One particular rule is called Hund s Rule in introductory chemistry courses. Hund s rule states that every orbital in a subshell is singly occupied with one electron before any one orbital is doubly occupied, and all electrons in singly occupied orbitals have the same spin. 

A sensitizer is an electronically excited species that transfers energy to a ground state molecule, leading to the ground electronic state of the sensitizer and an electronically excited state of the energy acceptor. The symbol means that the sensitizer is a triplet (i.e., the two electrons in singly occupied orbitals have the same spin). These terms will be discussed in Chapter 12. 

The explanation of Hund s rule is complicated, but it reflects the quantum mechanical property of spin correlation, that electrons in different orbitals with parallel spins have a quantum mechanical tendency to stay well apart (a tendency that has nothing to do with their charge even two uncharged electrons would behave in the same way). Their mutual avoidance allows the atom to shrink slightly, so the electron-nucleus interaction is improved when the spins are parallel. We can now conclude that in the ground state of a C atom, the two 2p electrons have the same spin, that all three 2p electrons in an N atom have the same spin, and that the two electrons that singly occupy different 2p orbitals in an O atom have the same spin (the two in the 2p , orbital are necessarily paired). 

Fig. 4.8 Exciton-breather, covalent channel, (K, K ), evolving from the excited state of a chain of 198 sites ground state geometry, but one electron excited from the highest level of the valence band to the lowest level of the conduction band. To break the symmetry, a small electric field is applied (potential drop to the right). Time evolution of S, (strobelight every 10 Ar). Time evolution of eigenvalues (both intergap states singly occupied) and of S, of central bonds.

In practice, each CSF is a Slater determinant of molecular orbitals, which are divided into three types inactive (doubly occupied), virtual (unoccupied), and active (variable occupancy). The active orbitals are used to build up the various CSFs, and so introduce flexibility into the wave function by including configurations that can describe different situations. Approximate electronic-state wave functions are then provided by the eigenfunctions of the electronic Flamiltonian in the CSF basis. This contrasts to standard FIF theory in which only a single determinant is used, without active orbitals. The use of CSFs, gives the MCSCF wave function a structure that can be interpreted using chemical pictures of electronic configurations [229]. An interpretation in terms of valence bond sti uctures has also been developed, which is very useful for description of a chemical process (see the appendix in [230] and references cited therein). 

If the mini her of electrons, N, is even, yon can haven dosed shell (as shown ) where the occupied orbitals each contain two electron s. For an odd n nrn her of electron s, at least on e orbital rn ust be singly occupied. In the example, three orbitals are occupied by-electron s and two orbitals arc nn occupied. Th e h ighest occupied nioleciilar orbital (HOMO is t[r), and the lowest unoccupied molecular orbital (LUMO) is The example above is a singlet, a state oh total spin S=0. Exciting one electron from the HOMO to the LUMO orbital would give one ol the I ollowing excited states ... 

A more general way to treat systems having an odd number of electrons, and certain electronically excited states of other systems, is to let the individual HF orbitals become singly occupied, as in Figure 6.3. In standard HF theory, we constrain the wavefunction so that every HF orbital is doubly occupied. The idea of unrestricted Hartree-Fock (UHF) theory is to allow the a and yS electrons to have different spatial wavefunctions. In the LCAO variant of UHF theory, we seek LCAO coefficients for the a spin and yS spin orbitals separately. These are determined from coupled matrix eigenvalue problems that are very similar to the closed-shell case. 

In Chapter 6, I discussed the open-shell HF-LCAO model. 1 considered the simple case where we had ti doubly occupied orbitals and 2 orbitals all singly occupied by parallel spin electrons. The ground-state wavefunction was a single Slater determinant. I explained that it was possible to derive an expression for the electronic energy... 

For diatomics with ten valence electrons, pole strengths lie between 0.86 and 0.89. DOs are dominated by a single occupied orbital in all cases. In the normalized DO for the state of AlO, there are other contributions with coefficients near 0.02. For the states of BO and AlO, certain operators have U elements that are approximately 0.1. Recent experimental work has produced a revised figure, 2.508 0.008 eV, for the electron affinity of BO [42] and the entry in Table III is in excellent agreement. Similar agreement occurs for the electron affinities of CN, AlO and AIS. 

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