Ground-state wave function electronic Hamiltonian, spin-orbit

The ground state wave function ([SKo)) of an JV-electron system with Hamiltonian Jfo is a determinant formed from the N spin orbitals with lowest energies,... 

We have placed special emphasis on using a consistent notation throughout the book. Since quantum chemists use a number of different notations, it is appropriate to define the notation we have adopted. Spatial molecular orbitals (with latin indices ij, k...) are denoted by These are usually expanded in a set of spatial (atomic) basis functions (with greek indices ju, V, A,...) denoted by 0. Molecular spin orbitals are denoted by x Occupied molecular orbitals are specifically labeled by a, b, c,... and unoccupied (virtual) molecular orbitals are specifically labeled by r, s, r,... Many-electron operators are denoted by capital script letters (for example, the Hamiltonian is Jf), and one-electron operators are denoted by lower case latin letters (for example, the Fock operator for electron-one is /( )). The exact many-electron wave function is denoted by O, and we use T to denote approximate many-electron wave functions (i.e., the Hartree-Fock ground state wave function is o while FS is a doubly excited wave function). Exact and approximate energies are denoted by S and , respectively. All numerical quantities (energies, dipole moments, etc.) are given in atomic units. 

Since H° is the sum of hydrogenlike Hamiltonians, the zeroth-order wave function is the product of hydrogenlike functions, one for each electron. We call any one-electron spatial wave function an orbital. To allow for electron spin, each spatial orbital is multiplied by a spin function (either a or 0) to give a spin-orbital. To introduce the required antisymmetry into the wave function, we take the zeroth-order wave function as a Slater determinant of spin-orbitals. For example, for the Li ground state, the normalized zeroth-order wave function is... 

The inversion operation i which leads to the g/u classification of the electronic states is not a true symmetry operation because it does not commute with the Fermi contact hyperfine Hamiltonian. The operator i acts within the molecule-fixed axis system on electron orbital and vibrational coordinates only. It does not affect electron or nuclear spin coordinates and therefore cannot be used to classify the total wave function of the molecule. Since g and u are not exact labels, it was realised by Bunker and Moss [265] that electric dipole pure rotational transitions were possible in ll], the g/u symmetry breaking (and simultaneous ortho-para mixing) being relatively large for levels very close to the dissociation asymptote. The electric dipole transition moment for the 19,1 19,0 rotational transition in the ground electronic state was calculated... 

The dominant diamagnetic Hamiltonian term is a simple one-electron operator and its expectation value, when the ground-state determinantal wave function is constructed from the set of occupied molecular spin-orbitals, is... 

A theoretical estimate of the spin spin parameter in the effective Hamiltonian can be made by the use of equations (7.111) and (7.121) using either ab initio or simpler, less accurate wave functions. This was done for the case of O2 in its ground state in an influential paper by Kayama and Baird [39]. They demonstrated that, in the case of a configuration (which occurs quite commonly in practice), there is significant spin-orbit mixing between the lowest Xl state and the metastable S+ state which arises from the same electron configmation. This has the effect of lowering the = 0 component of the state and so makes a positive contribution to the spin... 

We recall that the Hamiltonian, O Eq. 11.2, does not involve the electron spin. Spin-restricted implementations of electronic structure methods, therefore, only optimize the wave function parameters of the same spin symmetry as the reference wave function. Since operators that change the spin symmetry are known not to contribute to the ground-state energy (at least in the nonrelativistic picture in which spin-orbit effects are ignored), such spin-breaking wave function parameters are left imdefined. This has the consequence that average values of... 

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