Sum of two operators

The sum of two operators is an operator. Thus the Hamiltonian operator for the hydrogen atom has — j as the kinetic energy part owing to its single election plus — 1/r as the electiostatic potential energy part, because the charge on the nucleus is Z = 1, the force is atrtactive, and there is one election at a distance r from the nucleus... 

The dispersion contribution to the interaction energy in small molecular clusters has been extensively studied in the past decades. The expression used in PCM is based on the formulation of the theory expressed in terms of dynamical polarizabilities. The Qdis(r, r ) operator is reworked as the sum of two operators, mono- and bielectronic, both based on the solvent electronic charge distribution averaged over the whole body of the solvent. For the two-electron term there is the need for two properties of the solvent (its refractive index ns, and the first ionization potential) and for a property of the solute, the average transition energy toM. The two operators are inserted in the Hamiltonian (1.2) in the form of a discretized surface integral, with a finite number of elements [15]. 

Expressed in this way, the dipole moment operator is the sum of two operators, the first of which is the Hermitean conjugate of the other ... 

Although a mathematical operator is a symbol that stands for the carrying out of an operation, we can define an operator algebra in which we manipulate these symbols much as we manipulate variables and numbers in ordinary algebra. We define the sum of two operators by... 

Addition and multiplication of operators of this kind can be defined in a straightforward manner. The sum of two operators G, Gi is that operator associated with Gi(/,/ ) + G2(/,/ ) It is easy to show that addition in this sense obeys all the usual rules. Similarly, multiplication of an operator by a constant may be defined in an obvious manner. The zero operator is that corresponding to G(t, t ) equal to zero for all t, t ... 

We may develop an algebra of operators, just as we can develop an algebra of numbers. The sum of two operators a and p is defined by the equation... 

The non-adiabatic operator matrix, A can be written as a sum of two terms a matrix of numbers, G, and a derivative operator matrix... 

To apply S. or L. to aa 3pi3po, one must realize that eaeh of these operators is, in turn, a sum of lowering operators for eaeh of the two open-shell eleetrons ... 

Effective See Operative Environmental The sum of two-thirds of the mean radiant temperature and one-third of the air temperature. 

The unperturbed Hamiltonian operator is the sum of two hydrogen-like Hamiltonian operators, one for each electron... 

Here, A and B run over the M nuclei while i and j denote the N electrons in the system. The first two terms describe the kinetic energy of the electrons and nuclei respectively, where the Laplacian operator V2 is defined as a sum of differential operators (in cartesian coordinates)... 

Rather than splitting the physical space into short- and long-range parts as in the above techniques, an alternative is for the Coulomb operator itself to be reformulated and written as a sum of two contributions representing the short- and long-range regimes,... 

A position vector like p above is usually represented as starting at the origin, but it remains unchanged when moved to another position in the plane, as long as its length and direction do not change. Such an operation is used to form the sum of two vectors, A and B. By moving either A or B to start from the end point of the second vector, the same vector sum is obtained. It is easy to see how the components add up to... 

By postulating the correlation operator to be a sum of two-electron operators and assuming the occupied orbitals to be localized, we were able to show that the correlation energy can in fact be approximately expressed in terms of the bilinear expression... 

Problem 11-24. Write the function of problem 11-22, F x,y,z) = +y z +xz as the sum of two functions symmetric and antisymmetric with respect to (reflection in the x-z plane). Write a projection operator that would project out the component of a function symmetric with respect to ... 

This expression may be viewed as a sum of two terms. The first one, the integral over the squared derivative of the dipole function, p R), models those contributions which arise from the variation of the dipole strength with the separation R. The second models the contributions due to the variation of direction, p, as the rather natural separation of the kinetic energy operator into a radial and angular part suggests [314]. 

Operators corresponding to physical quantities, in second-quantization representation, are written in a very simple form. In the quantum mechanics of identical particles we normally have to deal with two types of operators symmetric in the coordinates of all particles. The first type includes N-particle operators that are the sum of one-particle operators. An example of such an operator is the Hamiltonian of a system of noninteracting electrons (e.g. the first two terms in (1.15)). The second type are iV-particle operators that are the sum of two-particle operators (e.g. the energy operator for the electrostatic interaction of electrons - the last term in (1.15)). In conventional representations these operators are... 

Nonadiabaticity—the additional "friction" between electrons and nuclei (ions)—neglected in obtaining the BO approximation is describable by an operator H. VJe now evaluate H (72,73) and estimate the accuracy of the BO adiabatic approximation. The operator H can be found in the following way (29). The total Hamiltonian H is written as a sum of two terms,... 

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