Hamiltonian operator purely electronic

The standard quantum chemical model for the molecular hamiltonian Hm contains, besides purely electronic terms, the Coulomb repulsion among the nuclei Vnn and the kinetic energy operator K]. The electronic terms are the electron kinetic energy operator Ke and the electron-electron Coulomb repulsion interaction Vee and interactions of electrons with the nuclei, these latter acting as sources of external (to the electrons) potential designated as Ve]q. The electronic hamiltonian He includes and is defined as... 

Extended Hiickel Theory (EHT) uses the highest degree of approximation of any of the approaches we have already considered. The Hamiltonian operator is the least complex and the basis set of orbitals includes only pure outer atomic orbitals for each atom in the molecule. Many of the interactions that would be considered in semi-empirical MO theory are ignored in EHT. EHT calculations are the least computationally expensive of all, which means that the method is often used as a quick and dirty means of obtaining electronic information about a molecule. EHT is suitable for all elements in the periodic table, so it may be applied to organometallic chemistry. Although molecular orbital energy values and thermodynamic information about a molecule are not accessible from EHT calculations, the method does provide useful information about the shape and contour of molecular orbitals. 

The first term is the electronic kinetic-energy operator the second and third terms are the attractions between the electron and the nuclei. In atomic units the purely electronic Hamiltonian for is... 

Without consulting the text, write down the complete nonrelativistic Hamiltonian operator for the H2 molecule. Then write down the purely electronic Hamiltonian operator for H2. 

The expression for the Hartree-Fock molecular electronic energy hf is given by the variation theorem as = D H i + FawI-D), where D is the Slater-determinant Hartree-Fock wave function, and the purely electronic Hamiltonian and the inter-nuclear repulsion are given by (13.5) and (13.6). Since doesn t involve electronic coordinates and D is normalized, we have D Vnn D) = Vnn D D) = The operator is the sum of one-electron operators /, and two electron operators gj/, we have gij, where (in atomic units)... 

There are useful two- and many-electron analogues of the functions discussed above, but when the Hamiltonian contains only one- and two-body operators it is sufficient to consider the pair functions thus the analogue of p(x x ) is the pair density matrix 7t(xi,X2 x i,x ) while that of which follows on identifying and integrating over spin variables as in (4), is H(ri,r2 r i,r2)- When the electron-electron interaction is purely coulombic, only the diagonal element H(ri,r2) is required and the expression for the total interaction energy becomes... 

There is a general statement [17] that spin-orbit interaction in ID systems with Aharonov-Bohm geometry produces additional reduction factors in the Fourier expansion of thermodynamic or transport quantities. This statement holds for spin-orbit Hamiltonians for which the transfer matrix is factorized into spin-orbit and spatial parts. In a pure ID case the spin-orbit interaction is represented by the Hamiltonian //= a so)pxaz, which is the product of spin-dependent and spatial operators, and thus it satisfies the above described requirements. However, as was shown by direct calculation in Ref. [4], spin-orbit interaction of electrons in ID quantum wires formed in 2DEG by an in-plane confinement potential can not be reduced to the Hamiltonian H s. Instead, a violation of left-right symmetry of ID electron transport, characterized by a dispersion asymmetry parameter Aa, appears. We show now that in quantum wires with broken chiral symmetry the spin-orbit interaction enhances persistent current. 

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... 

An important point to note is that B in the field-free Hamiltonian (11.7) is the rotational moment operator, and can have different values in (11.8) and (11.9). From these general expressions we can calculate the matrix elements involving the five primitive fimctions as shown below. In order to evaluate these matrix elements it is necessary to specify a value of L Freund and Miller pointed out that L = 2 for a d electron. This is the pure... 

The outline of the review is as follows in the next section (Sect. 2) we introduce the basic ideas of effective Hamiltonian theory based on the use of projection operators. The effective Hamiltonian (1-5) for the ligand field problem is constructed in several steps first by analogy with r-electron theory we use the group product function method of Lykos and Parr to define a set of n-electron wavefimctions which define a subspace of the full -particle Hilbert space in which we can give a detailed analysis of the Schrodinger equation for the full molecular Hamiltonian H (Sect. 3 and 4). This subspace consists of fully antisymmetrized product wavefimctions composed of a fixed ground state wavefunction, for the electrons in the molecule other than the electrons which are placed in states, constructed out of pure d-orbitals on the... 

The Hamiltonian is now a pure nuclear operator dictating, in the Born-Oppenheimer approximation, the evolution of the nuclear wavefunction [115]. The electrons enter Hn only through a potential energy term, o(R/), added to the bare nuclei-nuclei interaction Vnn- This potential energy term due to the electrons is the ground state energy of the electronic system at fixed ionic configuration. 

It is a fundamental fact of quantum mechanics, that a spin-independent Hamiltonian will have pure spin eigenstates. For approximate wave functions that do not fulfill this criterion, e.g. those obtained with various unrestricted methods, the expectation value of the square of the total spin angular momentum operator, (5 ), has been used as a measure of the degree of spin contamination. is obviously a two-electron operator and the evaluation of its expectation value thus requires knowledge of the two-electron density matrix. 

We have used the same symbol L for the transformed one-electron part of the Hamiltonian as in section 6, although there is a slight difference. In the pure one-electron case of sec. 6, the operator L was determined by the condition that its nondiagonal part vanishes, i.e. that it does not couple states within the model space with states outside of it. Now we cannot require, a priori, that the nondiagonal part of L vanishes. However, we decompose L into a contribution L that corresponds to its diagonal part, and a non-diagonal remainder, and make a similar decomposition of the two-electron operator G. We shall see that the non-diagonal remainders do not contribute to expectation values. 

As before, we stick to the one-electron case and leave the generalization to N electrons to the reader. The proper choice of the DKH expansion parameter, V or V(A), for the DKH transformation is a decisive question in the sequential unitary transformation scheme as it produces a series expansion of the block-diagonal operator with each term to be classified according to a well-defined order in the expansion parameter. In the case of a one-step decoupling scheme (for instance, in a purely numerical fashion as suggested by Barysz and Sadlej for the Hamiltonian see section 11.6) all derivations of this section are also valid. 

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