One-dimensional potential

For eaeh of the one-dimensional potential energy graphs shown below, determine ... [Pg.78]

A particle of mass m moves in a one-dimensional potential given by H = -----h... [Pg.87]

If all the PES coordinates are split off in this way, the original multidimensional problem reduces to that of one-dimensional tunneling in the effective barrier (1.10) of a particle which is coupled to the heat bath. This problem is known as the dissipative tunneling problem, which has been intensively studied for the past 15 years, primarily in connection with tunneling phenomena in solid state physics [Caldeira and Leggett 1983]. Interaction with the heat bath leads to the friction force that acts on the particle moving in the one-dimensional potential (1.10), and, as a consequence, a> is replaced by the Kramers frequency [Kramers 1940] defined by... [Pg.9]

This formula, however, tacitly supposes that the instanton period depends monotonically on its amplitude so that the zero-amplitude vibrations in the upside-down barrier possess the smallest possible period 2nla>. This is obvious for sufficiently nonpathological one-dimensional potentials, but in two dimensions this is not necessarily the case. Benderskii et al. [1993] have found that there are certain cases of strongly bent two-dimensional PES when the instanton period has a minimum at a finite amplitude. Therefore, the cross-over temperature, formally defined as the lowest temperature at which the instanton still exists, turns out to be higher than that predicted by (4.7). At 7 > Tc the trivial solution Q= Q Q is the saddle-point coordinate) emerges instead of instanton, the action equals S = pV (where F " is the barrier height at the saddle point) and the Arrhenius dependence k oc exp( — F ") holds. [Pg.61]

The dimensionless upside-down barrier frequency equals = 2(1 — and the transverse frequency Qf = Q. The instanton action at = oo in the one-dimensional potential (4.41) equals [cf. eq. (3.68)]... [Pg.71]

The total Hamiltonian is the sum of the two terms H = H + //osc- The way in which the rate constant is obtained from this Hamiltonian depends on whether the reaction is adiabatic or nonadiabatic, concepts that are explained in Fig. 2.2, which shows a simplified, one-dimensional potential energy surface for the reaction. In the absence of an electronic interaction between the reactant and the metal (i.e., all Vk = 0), there are two parabolic surfaces one for the initial state labeled A, and one for the final state B. In the presence of an electronic interaction, the two surfaces split at their intersection point. When a thermal fluctuation takes the system to the intersection, electron transfer can occur in this case, the system follows the path... [Pg.35]

Consider a particle of mass wr in a one-dimensional potential such that... [Pg.64]

To render the KP theory feasible for many-body systems with N particles, we make the approximation of independent instantaneous normal mode (INM) coordinates [qx° 3N for a given configuration xo 3W [12, 13], Hence the multidimensional V effectively reduces to 3N one-dimensional potentials along each normal mode coordinate. Note that INM are naturally decoupled through the 2nd order Taylor expansion. The INM approximation has also been used elsewhere. This approximation is particularly suited for the KP theory because of the exponential decaying property of the Gaussian convolution integrals in Eq. (4-26). The total effective centroid potential for N nuclei can be simplified as ... [Pg.92]

Only those problems that can be reduced to one-dimensional one-particle problems can be solved in closed form by the methods of wave mechanics, which excludes all systems of chemical interest. As shown before, several chemical systems can be approximated by one-dimensional model systems, such as a rotating diatomic molecule modelled in terms of a rotating particle in a fixed orbit. The trick is to find a one-dimensional potential function, V that provides an approximate model of the interaction of interest, in the Schrodinger formulation... [Pg.299]

Figure 5.12 depicts the corresponding adiabatic one-dimensional potential for the covalent rHF proton-transfer coordinate, showing the barrierless switch-over at equilibrium between FH F and F HF bond patterns. The potential well is seen to be extremely flat in the neighborhood of equilibrium, corresponding to the extremely low IR frequency of the proton-transfer mode (1299 cm-1, red-shifted... [Pg.620]

The reaction of hydrogen gas with a metal is called the absorption process and can be described in terms of a simplified one-dimensional potential energy curve (onedimensional Lennard-Jones potential) [30] (Figure 5.22). [Pg.130]

Lets use these ideas to solve some problems focusing our attention on the harmonic oscillator a particle of mass m moving in a one-dimensional potential described by V(x) =... [Pg.429]

The standard language used to describe rate phenomena in condensed phases has evolved from Kramers one dimensional model of a particle moving on a one dimensional potential, feeling a random and a related friction force. In Section II, we will review the classical Generalized Langevin Equation (GEE) underlying Kramers model and its application to condensed phase systems. The GLE has an equivalent Hamiltonian representation in terms of a particle which is bilinearly coupled to a harmonic bath. The Hamiltonian representation, also reviewed in Section II is the basis for a quantum representation of rate processes in condensed phases. Eas also been very useful in obtaining solutions to the classical GLE. Variational estimates for the classical reaction rate are described in Section III. [Pg.2]

For many one dimensional potentials, the infinite period trajectory is known analytically so that also the Fourier transformed force F(A) is known analytically. Finding the energy loss reduces then to a single quadrature. [Pg.18]

Figure 5.4 shows the one-dimensional potential V(x) of the Kronig-Penney model, which comprises square wells that are separated by barriers of height,... [Pg.112]

Figure 7.5 A (reasonably accurate) one-dimensional potential energy diagram for 238U indicating the energy and calculated distances for a decay into 234Th. Fermi energy Rs30 MeV, Coulomb barrier -28 MeV at 9.3 fm, Qa 4.2 MeV, distance of closest approach 62 fm. (Figure also appears in color figure section.)...

Another important quantum mechanical problem of interest to nuclear chemists is the penetration of a one-dimensional potential barrier by a beam of particles. The results of solving this problem (and more complicated variations of the problem) will be used in our study of nuclear a decay and nuclear reactions. The situation is shown in Figure E.5. A beam of particles originating at — oo is incident on a barrier of thickness L and height V0 that extends from x = 0 to x = L. Each particle has a total energy E. (Classically, we would expect if E < V0, the particles would bounce off the barrier, whereas if E > V0, the particles would pass by the barrier... [Pg.654]

The standard language used to describe rate phenomena in condensed phases has evolved from Kramers one dimensional model of a particle moving on a one dimensional potential, feeling a random and a related friction force.39 In Section... [Pg.2]

A hydrogen bond can be loosely described as the interaction between two electronegative atoms and an intervening hydrogen atom, i.e., -XH Y-, where X and Y represent O, F, Cl, N, S, or C atoms. The potential surface that describes this interaction typically has two minima that correspond to formation of a strong XH or YH bonds. A one-dimensional potential that describes the motion of the central H atom between the two heavier atoms at a fixed distance R from each other can be represented by the empirical form... [Pg.152]

Big Chemical Encyclopedia

Chemical substances, components, reactions, process design ...