Open-shell wavefunction

The organization is transparent to the type of parameter provided that the derivative Fock operators are suitably constructed. The separation of terms in Eqn. (69) mentioned here corresponds to the idea of Takada, Dupuis, and King for skeleton Fock matrices [64]. A geometric derivative can be obtained with the same code as an electrical property, as can mixed derivatives. Furthermore, magnetic properties, which are unique because second derivatives of the Hamiltonian operator are not necessarily zero, fit into the DHF structure without modification. Open-shell wavefunctions can be em-... 

One deficiency can be seen in the case of a high-spin open-shell wavefunction. If the condition that f should not be symmetry breaking is relaxed, by replacing D with the spin density matrix and using spin orbitals, we arrive at... 

SCF, the only significant change will be that the CHF equations (33) and (34) will be different. The relevant equations for UHF have been given by Pople et for high-spin open-shell wavefunctions the theory has been examined by Saxe et and Osamura et al. have considered general open-shell... 

As a somewhat more complex example, let us now consider the case of ozone (O3), which has an open-shell singlet ground state (Sidebar 3.2). The Gaussian input file to obtain the open-shell wavefunction and default NBO analysis for experimental equilibrium geometry Roo= 1-272, 0= 116.8 ) is shown below. 

The fact is that the molecular orbitals describing the resulting cation may well be quite different from those of the parent molecule. We speak of electron relaxation, and so we need to examine the problem of calculating accurate HF wavefunctions for open-shell systems. 

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... 

It is possible to construct a HF method for open-shell molecules that does maintain the proper spin symmetry. It is known as the restricted open-shell HF (ROHF) method. Rather than dividing the electrons into spin-up and spin-down classes, the ROHF method partitions the electrons into closed- and open-shell. In the easiest case of the high-spin wavefunction ( op = — electrons in op... 

Nakatsuji H, Hirao K (1978) Cluster expansion of the wavefunction. symmetry-adapted-cluster expansion, its variational determination, and extension of open-shell orbital theory. J Chem Phys 68 2053... 

Although the above discussion assumes that all MOs are occupied by two electrons, it turns out that the basic ideas can be extended to open-shell molecules in which there are unequal numbers of electrons in the two spin states. Without showing the complicated mathematics, we will show how the wavefunction can be determined by constructing two Fock matrices for each spin state and then solving two sets of coupled Roothaan equations ... 

To distinguish between closed-shell and open-shell configurations (and determinants), one may generally include a prefix to specify whether the starting HF wavefunction is of restricted closed-shell (R), restricted open-shell (RO), or unrestricted (U) form. (The restricted forms are total S2 spin eigenfunctions, but the unrestricted form need not be.) Thus, the abbreviations RHF, ROHF, and UHF refer to the spin-restricted closed-shell, spin-restricted open-shell, and unrestricted HF methods, respectively. 

Perturbative approximation methods are usually based on the Mpller-Plesset (MP) perturbation theory for correcting the HF wavefunction. Energetic corrections may be calculated to second (MP2), third (MP3), or higher order. As usual, the open- versus closed-shell character of the wavefunction can be specified by an appropriate prefix, such as ROMP2 or UMP2 for restricted open-shell or unrestricted MP2, respectively. 

As explained in the Introduction, this scheme breaks down both for ground and excited states when orbitals from occupied and virtual subspaces become near-degenerate, e.g. at the dissociation limit or in diradicals (see Figure 2). To overcome this problem, the SF model employs a high-spin triplet reference state which is accurately described by a SR wavefunction. The target states, closed and open shell singlets and triplets, are described as spin-flipping excitations ... 

Therefore, the SF ansatz (2) is sufficiently flexible to describe changes in ground state wavefunctions along a single bond-breaking coordinate. Moreover, it treats both closed-shell (e.g., N and Z) and open-shell (V and T) diradicals states in a balanced fashion, i.e., without overemphasizing the importance of one of the configurations. 

The lowest-lying n=>7t states correspond to a configuration (only those orbitals whose occupancies differ from those of the ground state are listed) of the form 2b212bi1, which gives rise to 1A2 and 2A2 wavefunctions (the direct product of the open-shell spin... 

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. 

The simplest approximate wavefunction for an open-shell molecule is the spin-unrestricted Hartree-Fock function... 

Open-shell Pseudohamiltonians.—The majority of atoms do not have valence structures which can be represented by the fully closed-shell wavefunction of equation (14), and consequently ab initio pseudopotentials cannot be derived directly from the theory outlined above. Acceptable wavefunctions for such atoms require either more than one determinant or the use of the symmetry-equivalenced or generalized Hartree-Fock method, and usually include partially filled shells. The total all-electron wavefunction may be symbolically expressed in terms of four subspaces,... 

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