Ehrenfest potential

The potential V(r) is closely related to the average potential defined by Slater [59, 60] we call it the Ehrenfest potential. For Ehrenfest potentials that are homogeneous of degree minus one in r, (16) gives the usual Coulombic virial relation between the total potential energy and the total kinetic energy, —V=2T. 

One can perform a QTAIM-Uke analysis based on the Ehrenfest force, since F(r) also defines a vector field with a (divergent) maximum at the nucleus. For any given point in space, Tq, one can follow the force-ascent lines to a nucleus, and decide that this atom exerts more force on the point Fq than the other atoms in the system. One must be cautious, however, because of the occasional presence of nonnuclear minima in the Ehrenfest potential [9, 10]. That is, in the same way that one may draw analogies between the stress tensor and the second derivative of the density, ff (r) VV p(r), one may draw analogies between the Ehrenfest force and the... 

The center of the wavepacket thus evolves along the trajectory defined by classical mechanics. This is in fact a general result for wavepackets in a hannonic potential, and follows from the Ehrenfest theorem [147] [see Eqs. (154,155) in Appendix C]. The equations of motion are straightforward to integrate, with the exception of the width matrix, Eq. (44). This equation is numerically unstable, and has been found to cause problems in practical applications using Morse potentials [148]. As a result, Heller inboduced the P-Z method as an alternative propagation method [24]. In this, the matrix A, is rewritten as a product of matrices... 

Both the BO dynamics and Gaussian wavepacket methods described above in Section n separate the nuclear and electronic motion at the outset, and use the concept of potential energy surfaces. In what is generally known as the Ehrenfest dynamics method, the picture is still of semiclassical nuclei and quantum mechanical electrons, but in a fundamentally different approach the electronic wave function is propagated at the same time as the pseudoparticles. These are driven by standard classical equations of motion, with the force provided by an instantaneous potential energy function... 

According to the correspondence principle as stated by N. Bohr (1928), the average behavior of a well-defined wave packet should agree with the classical-mechanical laws of motion for the particle that it represents. Thus, the expectation values of dynamical variables such as position, velocity, momentum, kinetic energy, potential energy, and force as calculated in quantum mechanics should obey the same relationships that the dynamical variables obey in classical theory. This feature of wave mechanics is illustrated by the derivation of two relationships known as Ehrenfest s theorems. 

Ehrenfest P (1933) Phase changes in the ordinary and extended sense classified according to the corresponding singularities of the thermodynamic potential. Proc Acad Sci Amsterdam 36 153-157... 

Ehrenfest, P. Phase changes classified according to the singularities of the thermodynamic potential. Proc. Acad. Sci., Amsterdam 36, 153 (1933) Suppl. 75b, Mitt. Kammerlingh Onnes Inst., Leiden... 

To differentiate between the variety of phase equilibria that occur, Ehrenfest proposed a classification of phase transitions based upon the behavior of the chemical potential of the system as it passed through the phase transition. He introduced the notion of an th order transition as one in which the nth derivative of the chemical potential with respect to T or p showed a discontinuity at the transition temperature. While modern theories of phase transitions have shown that the classification scheme fails at orders higher than one, Ehrenfest s nomenclature is still widely used by many scientists. We will review it here and give a brief account of its limitations. 

Fig. 3.2. Two-dimensional potential energy surface V(R, 7) (dashed contours) for the photodissociation of C1CN, calculated by Waite and Dunlap (1986) the energies are given in eV. The closed contours represent the total dissociation wavefunction tot R,l E) defined in analogy to (2.70) in Section 2.5 for the vibrational problem. The energy in the excited state is Ef = 2.133 eV. The heavy arrow illustrates a classical trajectory starting at the maximum of the wavefunction and having the same total energy as in the quantum mechanical calculation. The remarkable coincidence of the trajectory with the center of the wavefunction elucidates Ehrenfest s theorem (Cohen-Tannoudji, Diu, and Laloe 1977 ch.III). Reprinted from Schinke (1990).

Ehrenfest, P. Phasenumwandlungen im flblichen und erweiterten Sinn, klassifizicrt nach den entsprechenden Singularitaten des thermodynamischen Potentials. Leiden Comm. Suppl. 75 b, 8—13 (1933) Proc. Kon. Akad. Amsterdam 36, 153 (1933). 

In most of the more recent classical approaches [18], no allusion to Ehrenfest s (adiabatic) principle is employed, but rather the differential equations of motion from classical mechanics are solved, either exactly or approximately, subject to a set of initial conditions (masses, force constants, interaction potential, phase, and initial energies). The amount of energy, AE, transferred to the oscillator is obtained for these conditions. This quantity may then be averaged over all phases of the oscillating molecule. In approximate classical and semiclassical treatments, the interaction potential is expanded in a Taylor s series and only the first two terms are retained. 

The individual contributions to the electronic potential energy of an atom are obtained by determining the action of the operator — r-V on V in the expression for the basin virial, eqn (6.19). The basin virial is the viried of the Ehrenfest force and the force density F(r) of eqn (6.16) is evaluated as the first... 

Averaging of this final operator expression for the virial of V in the manner indicated in eqn (6.69) for the potential energy density TFbWi which is the virial of the Ehrenfest force eqn (6.29), yields... 

Electrostatic potential maps have been used to make predictions similar to these (Scrocco and Tomasi 1978). Such maps, however, do not in general reveal the location of the sites of nucleophilic attack (Politzer et al. 1982), as the maps are determined by only the classical part of the potential. The local virial theorem, eqn (7.4), determines the sign of the Laplacian of the charge density. The potential energy density -f (r) (eqn (6.30)) appearing in eqn (7.4) involves the full quantum potential. It contains the virial of the Ehrenfest force (eqn (6.29)), the force exerted on the electronic charge at a point in space (eqns (6.16) and (6.17)). The classical electrostatic force is one component of this total force. 

We have resorted to an approximate technique which attempts to include the above mentioned main quantum effects via the construction of effective potentials V. Basically, each pmticle is represented by a single particle wavefunction tmd the Ehrenfest theorem is applied. Similar ideas have been used with good success ev( n for quantum solids like hydrogmi [38]. Effective quantum potentials ajx also among the results of the Feynman-Hibbs treatment [12] which have been apjjlied to pure neon clusters in the past [34]. 

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