Hamiltonian electronic, eigenfunctions

Because the Hamiltonian operator defined by Eq. (4.32) is separable, its many-electron eigenfunctions can be constructed as products of one-electron eigenfunctions. That is... 

The procedure leading from the exact /V-electron Hamiltonian (2) to the Heisenberg Hamiltonian matrix (47) is very instructive, but it is rather lengthy. Much simpler is the use of effective Hamiltonians which in the space of N-electron eigenfunctions of S2 and S2 are represented by the same matrix. Furthermore, using the effective Hamiltonians may bring another insight into the nature of the interactions described by the model. The simplest effective Hamiltonian in the pure spin is... 

The electronic eigenfunctions of this Hamiltonian are Xo, , A where f o is the ground state, which we presume does not possess any electronic angular momentum, and in- are the excited states, which may possess electronic angular momentum. 

As far as electron-electron interaction is neglected, the Hamiltonian // a tt electron in a CNT commutes with all the element of G making, according to the basic theory of group representation [22], the electronic eigenfunctions a set of basis functions for the Irreducible representations of G. In fact, the basis functions = , of an irreducible representation of dimension I are characterized by the property... 

Of course, we are interested in solving the Schrodinger equation for the total Hamiltonian in Eq. (1) describing the electronic plus nuclear motion in our system. To this end, we expand the total wavefunction in the electronic eigenfunctions of Hf, ... 

The time-independent Schrodinger equation (SE) for a molecular system derives from Hamiltonian classical dynamics and includes atomic nuclei as well as electrons. Eigenfunctions are therefore functions of both electronic and nuclear coordinates. Very often, however, the nuclear and electronic variables can be separated. The motion of the heavy particles may be treated using classical mechanics. Particularly at high temperatures, the Heisenberg uncertainty relation Ap Ax > /i/2 is easy to satisfy for atomic nuclei, which have a particle mass at least 1836 times the electron mass. The immediate problem for us is to obtain a time-independent SE including not only the electrons but also the nuclei and subsequently solve the separation problem. 

N-electron eigenfunctions of the no-pair DCB Hamiltonian are approximated by a linear combination of M configmation-state functions,... 

The one-electron operators in the resulting approximate hamiltonian for an atom are hydrogenlike ion hamiltonians. Their eigenfunctions are called atomic orbitals. 

The electronic Hamiltonian and the comesponding eigenfunctions and eigenvalues are independent of the orientation of the nuclear body-fixed frame with respect to the space-fixed one, and hence depend only on m. The index i in Eq. (9) can span both discrete and continuous values. The q ) form... 

In a diabatic representation, the electronic wave functions are no longer eigenfunctions of the electronic Hamiltonian. The aim is instead that the functions are so chosen that the (nonlocal) non-adiabatic coupling operator matrix, A in Eq. (52), vanishes, and the couplings are represented by (local) potential operators. The nuclear Schrddinger equation is then written... 

In this work, relativistic effects are included in the no-pah or large component only approximation [13]. The total electronic Hamiltonian is H (r R) = H (r R) + H (r R), where H (r R) is the nom-elativistic Coulomb Hamiltonian and R) is a spin-orbit Hamiltonian. The relativistic (nomelativistic) eigenstates, are eigenfunctions of R)(H (r R)). Lower (upper)... 

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