Operators kinetic energy, definition

It follows from the definition of the functionals and Ti that the exchange-correlation functional is invariant under the rotation and translation of the electron density. This follows directly from the fact that the kinetic energy operator f and the two-electron operator W are invariant with respect to these transformations. This also has implications for the exchange-correlation functional. 

Let be the Hamiltonian of the system T the kinetic energy operator, U and / potential energy operators which are nonnegative definite such that ... 

Here, Vf is the kinetic energy operator for particle i, (with h = 1, 2m, = 1, for all i), Vi represents the interaction of particle i with an external potential, such as that associated with nuclei in the system, and Vij represents the mutual interaction between particles i and j. The definition of the singleparticle Hamiltonian, hi, is evident. We are interested in the solutions of the many-particle time-independent Schrodinger equation,... 

Next we can dispose of the matrix elements of the one-electron operator Hy,Eq. (3-3 b) the kinetic energy operator, the electron-nuclear attraction potential arising from the metal nucleus, and the spin-independent relativistic terms have spherical symmetry and can be treated through the definition of the basis orbitals ip, Eq. (3-11). The spin-... 

The definitions of the terms can be found in [30], but it is sufficient here to note that Ka represents the vibrational kinetic energy operator, Ep the electronic energy, Vn the nuclear repulsion operator, and the terms b and bo are elements of a matrix closely related to the inverse of the instantaneous inertia operator matrix. It should also be noted that the y terms arise from the interaction of the rotational with the electronic motion and tend to couple electronic states, even those diagonal in k. 

According to Lowdin s definition [9, 10] A system of electrons and atomic nuclei is said to form a molecule if the Coulombic Hamiltonian H —with the center of mass motion removed—has a discrete groimd-state energy Eq (see also [11-13] and references therein) where the total Hamiltonian H = H = He + T n + Uim is, respectively, the sum of the electronic Hamiltonian operator, the nuclear kinetic energy operator, and the nuclear-nuclear Coulomb interaction energy operator. Consider, within the Bom-Oppenheimer approximation, the electronic Hamiltonian operator (in the atomic units) of M ... 

For definiteness, we consider the case that for an (n+1)-dimensional problem, the variable which we separate from the others is a distance p from a point in an (n+1)-dimensional space [2l]. So, p is the hyperradius of an n-dimensional sphere, and will denote the remaining n coordinates (typically, hyperangles). Such a representation for the many body problem is being actively explored because the kinetic energy operator becomes essentially a Laplacian of a hyperspace. Also, p has proven to be a good choice as a nearly separable variable for many problems (see Section 3) other choices are of course possible, but may lead to the appearance of additional coupling terms in the following equations. 

In the traditional approach, the center of mass motion of the molecule is explicitly separated off, and nuclear coordinates are introduced which suitably describe molecular vibrations and rotations. The procedure is not unique and the resulting kinetic energy operators depend on the choice of coordinates made. We follow here the transparent analysis of Mead which is briefly sketched below. The center of mass coordinate is defined as usual, and the nuclear relative coordinates are introduced one at a time. Starting with an arbitrary chosen nucleus, the new coordinate for each nucleus is defined relative to the center of mass of the nuclei already introduced. This definition of relative nuclear coordinates, which we denote by Rq, leads to a particularly simple form of the kinetic energy operators (Jacobi coordinates for a discussion of suitable nuclear coordinates, see Ref. 51). The relative electronic coordinates r are defined, as usual, relative to the center of mass of the nuclei. 

The description of rotational motion is naturally performed in spherical coordinates. The two angular variables of rotational motion are generalized coordinates, free of additional constraints, as introduced in chapter 2. The transformation to spherical coordinates affects the definition of the angular momentum (operator) and subsequently the squared angular momentum (operator) which enters the kinetic energy (operator) expression. In order to avoid lengthy coordinate transformations of the latter containing second derivatives with respect to Cartesian coordinates, we may consider the situation in classical mechanics first and subsequently apply the correspondence principle. 

With the definition of a radial momentum in Eq. (4.108), we can now write the kinetic energy operator in spherical coordinates as... 

Using the original definition of the momentum operator and the classical form of kinetic energy, derive the onedimensional kinetic energy operator... 

It is instructive to compare the 2-component Hamiltonian T-Cl with the Pauli Hamiltonian, which was derived in the previous section. The term Ep contains the relativistic free-particle energy, which is well-behaved for all values of the momentum. Ep, which is the kinetic energy operator for the positive-energy states, is a positive definite operator. In the Pauli Hamiltonian we see that this operator is expanded in powers of p/mc, which does not converge if p/mc > 1—a situation that will occur in any potential if the electron is sufficiently close to the nucleus. As mentioned above, the mass-velocity term is not bounded from below and so cannot be used variationally. 

V(q) is assumed here to be the electronic ground state, for definiteness. If excited electronic states are of interest, one can introduce excited state electronic potentials V (q), V"(q), etc. The complete zero order solution is finally given by equation (17) with the kinetic energy operator T, along the reaction coordinate ... 

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. 

Here //co,c(l) has been dissected into a kinetic energy integral T and two potential energy integrals, F(H) and F(He). From the definition of the operator //co,c (Eq. 5.64 = 5.19) and the Roothaan-Hall expression for the integral //core (Eq. 5.79) we see that (the (1) emphasizes that these integrals involve the coordinates of only one electron) ... 

The averaging operation for the liquid droplet velocity described in the previous section introduces a particle velocity deviation from the mean (or correlated) velocity, noted as m" = Up — ui, and named the random uncorrelated velocity [280]. By definition, the statistical average (based on the particle probability density function) of this uncorrelated velocity is zero < u" >= 0. A conservation equation can be written for the associated kinetic energy 59i =< Up pip > /2 ... 

When r = r, eqn (El.2) becomes eqn (1.11) hence, p(r) is said to be a diagonal element of r< (r, r ). While eqns (1.11) and (El.2) are formally alike, one can calculate the kinetic energy from the latter but not from the former, for only in the latter can one insert the operator between the natural orbitals and let it act separately on or rjf. The average value of a two-electron property can be expressed in terms of the diagonal elements of the second-order density matrix r (ri,r2). Assuming a summation over electron spins, its definition is... 

The mathematical issues relevant to the definition of density functional derivatives can be considered in the simple model of noninteracting electrons. As in the KSC [4], this singles out the kinetic energy. The /V-electron Hamiltonian operator is H = T + V. Orbital functional derivatives determine the noninteracting OEL equations... 

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