Differentiation operator

Here, the kinetic operators are typically given by the corresponding Lapla-cians Tg = Ag and 7 = or similar selfadjoint differential operators. The corresponding solution = (x, q, t) describes what we call the full QD evolution of the system. 

The vector differential operator del (or nahla) written as V is defined by... 

Notiee that H is a seeond order differential operator in the spaee of the thirty-nine eartesian eoordinates that deseribe the positions of the ten eleetrons and three nuelei. It is a seeond order operator beeause the momenta appear in the kinetie energy as pj and pa, and the quantum meehanieal operator for eaeh momentum p = -ih d/dq is of first order. 

The differential operator terms ean be reeast in several ways using... 

By substituting /= F(r)/r into this equation, one obtains an equation for F(r) in whieh the differential operators appear to be less eomplieated ... 

The full Hamiltonian H thus eontains differential operators over the 3N eleetronie eoordinates (denoted r as a shorthand) and the 3M nuelear eoordinates (denoted R as a shorthand). In eontrast, the eleetronie Hamiltonian Hg is a Hermitian differential operator in... 

Although in this appendix, we choose to treat it as a collection of differential operators that... 

Dijferential Operations The following differential operations are valid /, g, are differentiable functions of x, c is a constant e is the base of the natural logarithms. 

Once the instanton trajectory has been numerically found, one proceeds to the calculation of prefactor, which amounts to finding determinants of differential operators. The direct two-dimensional generalization of (3.46) is... 

The differential operator on the left side of the equation is known as del-squared. The operator del is equivalent to partial differentiation with respect to x, y, and z components ... 

The nuclear kinetic energy is essentially a differential operator, and we may write it as ... 

The components of the operator P are hermitian.2 In general, any differential operator Q has a hermitian adjoint Qf, defined by the integral relation... 

But P(u) is not a function of t if the differential operator P is not an explicit function of t, which we shall assume here. Then... 

In this equation W and V are the energy constant and the potential energy, respectively the indicated partial differential operations involve coordinates whose line element is... 

There is strong evidence that DNA adduction by these bulky reactive metabolites of PAHs is far from random, and that there are certain hot spots that are preferentially attacked. Differential steric hindrance and the differential operation of DNA repair mechanisms ensure that particular sites on DNA are subject to stable adduct formation (Purchase 1994). DNA repair mechanisms clearly remove many PAH/ guanine adducts very quickly, but studies with P postlabeling have shown that certain adducts can be very persistent—certainly over many weeks. Evidence for this has been produced in studies on fish and Xenopus (an amphibian Reichert et al. 1991 Waters et al. 1994). 

After preliminary discussions of the simplest examples illustrating some ways of producing grids and, thereby, of forming the spaces Hh of grid functions we concentrate primarily on the problem of the difference approximation of differential operators. 

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