Hamiltonian electron kinetic energy

Here, t is the nuclear kinetic energy operator, and so all terms describing the electronic kinetic energy, electron-electron and electron-nuclear interactions, as well as the nuclear-nuclear interaction potential function, are collected together. This sum of terms is often called the clamped nuclei Hamiltonian as it describes the electrons moving around the nuclei at a particular configrrration R. 

The individual terms in (5.2) and (5.3) represent the nuclear-nuclear repulsion, the electronic kinetic energy, the electron-nuclear attraction, and the electron-electron repulsion, respectively. Thus, the BO Hamiltonian is of treacherous simplicity it merely contains the pairwise electrostatic interactions between the charged particles together with the kinetic energy of the electrons. Yet, the BO Hamiltonian provides a highly accurate description of molecules. Unless very heavy elements are involved, the exact solutions of the BO Hamiltonian allows for the prediction of molecular phenomena with spectroscopic accuracy that is... 

For a many-electron molecule, the Hamiltonian operator can thus be written as the sum of the electrons kinetic energy term, which in turn is the sum of individual electrons ... 

Quantum numbers and shapes of atomic orbitals Let us denote the one-electron hydrogenic Hamiltonian operator by h, to distinguish it from the many-electron H used elsewhere in this book. This operator contains terms to represent the electronic kinetic energy ( e) and potential energy of attraction to the nucleus (vne),... 

Ri,R2,. ..,Rk denotes the nuclear coordinates. The first two terms in equation (1) describe, respectively, the electronic kinetic energy and electron-nuclear attraction and the third term is a two-electron operator that represents the electron-electron repulsion. These three operators comprise the electronic Hamiltonian in free space. The term V(r) is a generic operator for an external potential. One of the common ways to express V(f), when it is affecting electrons only, is to expand it as a sum of one-electron contributions... 

For the electro-nuclear model, it is the charge the only homogeneous element between electron and nuclear states. The electronic part corresponds to fermion states, each one represented by a 2-spinor and a space part. Thus, it has always been natural to use the Coulomb Hamiltonian Hc(q,Q) as an entity to work with. The operator includes the electronic kinetic energy (Ke) and all electrostatic interaction operators (Vee + VeN + Vnn)- In fact this is a key operator for describing molecular physics events [1-3]. Let us consider the electronic space problem first exact solutions exist for this problem the wavefunctions are defined as /(q) do not mix up these functions with the previous electro-nuclear wavefunctions. At this level. He and S (total electronic spin operator) commute the spin operator appears in the kinematic operator V and H commute with the total angular momentum J=L+S in the I-ffame L is the total orbital angular momentum, the system is referred to a unique origin. 

Let us examine the Schrodinger equation in the context of a one-electron Hamiltonian a little more carefully. When the only terms in the Hamiltonian are the one-electron kinetic energy and nuclear attraction terms, the operator is separable and may be expressed as... 

Here h are the one-electron integrals including the electron kinetic energy and the electron-nuclear attraction terms, and gjjkl are the two-electron repulsion integrals defmed by (3 19). The summations in (3 24) are over the molecular orbital basis, and the definition is, of course, only valid as long as we work in this basis. Notice that the number of electrons does not appear in the defmition of the Hamiltonian. All such information is found in the Slater determinant basis. This is true for all operators in the second quantization formalism. 

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... 

Just like the helium Hamiltonian, the molecular Hamiltonian H in Eq. 5.15 is composed (from left to right) of electron kinetic energy terms, nucleus-electron... 

This is an important result. The first term leads to the rotational eigenvalues, whilst the second term describes the rotational electronic coupling and, as we shall see, contributes to the rotational magnetic moment and the spin rotation interaction. The third term is small and can be neglected for states where A = 0. We have omitted the electron kinetic energy term from (8.101) because it is part of the zeroth-order Hamiltonian which determines the electronic eigenvalues and eigenfunctions. 

Further development of Sommerfeld s theory of metals would extend well outside the intended scope of this textbook. The interested reader may refer to any of several books for this (e.g. Seitz, 1940). Rather, this book will discuss the band approximation based upon the Bloch scheme. In the Bloch scheme, Sommerfeld s model corresponds to an empty lattice, in which the electronic Hamiltonian contains only the electron kinetic-energy term. The lattice potential is assumed constant, and taken to be zero, without any loss of generality. The solutions of the time-independent Schrodinger equation in this case can be written as simple plane waves, = exp[/A r]. As the wave function does not change if one adds an arbitrary reciprocal-lattice vector, G, to the wave vector, k, BZ symmetry may be superimposed on the plane waves to reduce the number of wave vectors that must be considered ... 

It is conventional that the ligand field problem for systems with Na> d electrons requires the diagonalization of an effective Hamiltonian operator composed for the electronic kinetic energy T, and both one-electron ligand field terms, and two-electron Coulomb interactions ... 

In the course of differentiation of the Hamiltonian of Eq. (310) by components of R, all the electronic kinetic energy terms and all the terms describing electron-electron interactions are eliminated. Consequently, the force operator F is a one-electron operator and the expectation value can be written as... 

Second, the Hamiltonian operator for a relativistic many-body system does not have the simple, well-known form of that for the non-relativistic formulation, i.e. a sum of a sum of one-electron operators, describing the electronic kinetic energy and the electron-nucleus interactions, and a sum of two-electron terms associated with the Coulomb repulsion between the electrons. The relativistic many-electron Hamiltonian cannot be written in closed form it may be derived perturbatively from quantum electrodynamics.46... 

The electronic kinetic energy operator (25) simplifies and, adding the electron repulsion term, the electronic Hamiltonian is... 

The configuration coordinates of electrons (p) and nuclei (R) in the new frame are related to the laboratory one by rk = u + pk, Qk. = u + Rk, symbolically written as r =u+p, Q=u+R, and T=(p,R). Ke represents the electrons kinetic energy operators Vee (p), VeN(p, R) and Vnn(R) are the standard Coulomb interaction potentials they are invariant to origin translation. The vector u is just a vector in real space R3. Kn is the kinetic energy operator of the nuclei, and in this work the electronic Hamiltonian He(r Z) includes all Coulomb interactions. This Hamiltonian would represent a general electronic system submitted to arbitrary sources of external Coulomb potential. 

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