Kinetic energy operator, transformed, with

Hence, the method of Mead and Truhlar [6] yields a single-valued nuclear wave function by adding a vector potential A to the kinetic energy operator. Different values of odd (or even) I yield physically equivalent results, since they yield (< )) that are identical to within an integer number of factors of exp(/< )). By analogy with electromagnetic vector potentials, one can say that different odd (or even) I are related by a gauge transformation [6, 7]. 

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. 

This expansion is valid to second order with respect to St. This is a convenient and practical method for computing the propagation of a wave packet. The computation consists of multiplying X t)) by three exponential operators. In the first step, the wave packet at time t in the coordinate representation is simply multiplied by the first exponential operator, because this operator is also expressed in coordinate space. In the second step, the wave packet is transformed into momentum space by a fast Fourier transform. The result is then multiplied by the middle exponential function containing the kinetic energy operator. In the third step, the wave packet is transformed back into coordinate space and multiplied by the remaining exponential operator, which again contains the potential. 

Fourier transforms are often used in connection with periodic functions, for example for evaluating the kinetic energy operator in a density functional calculation where the orbitals are expanded in a plane wave basis. 

The Schrodinger equation is invariant with respeet to the Galilean transformation. Indeed, the Hamiltonian contains the potential energy, which depends on interparticle distances (i.e., on the differences of the coordinates), whereas the kinetic energy operator contains the second derivative operators that are invariant with respect to the Galilean transformation. Also, since t = t, the time derivative in the time-dependent Schrodinger equation does not change. 

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. 

In one dimension and with a complete set of electronic states, such a transformation is always possible. Tjv becomes diagonal, and V R) now has off-diagonal elements. The off-diagonal elements are simple functions and not operators so that they are much easier to make use of. This is the di-abatic representation, in which we now have diabatic electronic states. In the case of more than one dimension or with an incomplete set of electronic states, this transformation is not exactly possible. But frequently (or we are hoping that) a transformation which makes the off-diagonal kinetic energy operator matrix elements almost zero is possible. This is also named the diabatic representation (or the quasi-diabatic representation when we have need to make clear that the transformation is not exact). 

This kinetic energy operator treats all the motions of the two particles (the electron and the nucleus). We re specifically interested in the motion of the electron relative to the nucleus, but not the overall motion of the whole atom from, say, one side of the room to the other side of the room. A common transformation in physics (Eq. A.45) allows us to rearrange the kinetic energy terms for the nucleus and the electron into a reduced mass term for the relative motions, with the reduced mass jU = + m ), and a center o/moss term for the over-... 

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