Momentum distribution

It does not require knowledge of tlie factor nonnalizing tlie 6, i.e., the partition fiinction. For atomic and molecular systems, the partition fiinction is split into a product of ideal (exactly calculable) and excess tenns tlie position and momentum distributions also factorize, and we wish to sample... 

The denominator in Eq. (13) can be interpreted as an average value over the momentum distribution from the initial wavepacket, that is,... 

Is the temperature 1/0 related to the variance of the momentum distribution as in the classical equipartition theorem It happens that there is no simple generalization of the equipartition theorem of classical statistical mechanics. For the 2N dimensional phase space F = (xi. .. XN,pi,.. -Pn) the ensemble average for a harmonic system is... 

Crystal can compute a number of properties, such as Mulliken population analysis, electron density, multipoles. X-ray structure factors, electrostatic potential, band structures, Fermi contact densities, hyperfine tensors, DOS, electron momentum distribution, and Compton profiles. 

Mach s principle, as formulated by Wheeler [wheel64a], states that the inertial properties of an object are determined by the energy-momentum distribution throughout all of space. 

Fig. 5.1. The semiclassical description of angular momentum distribution relaxation (a) and rotational energy relaxation at Ea — 0 (6).

The momentum distribution, i.e. the diagonal element of (III. 10) then satisfies... 

The reciprocal form factor [22] is the Fourier transform of the momentum distribution,... 

We notice that neither the momentum distribution nor the reciprocal form factor seems to carry any information about the translational part of the space group. The non diagonal elements of the number density matrix in momentum space, on the other hand, transform under the elements of the space group in a way which brings in the translational parts explicitly. 

Lundqvist, B.I. and Lyden, C. (1971) Calculated momentum distributions and Compton profiles of interacting conduction electrons in lithium and sodium, Phys. Rev., B4, 3360-3370. 

Figure 3. Comparison of the measured momentum distributions of the outermost valence orbital for wafer [6-8] with spherically averaged orbital densities from Hartree-Fock limit and correlated wave functions [6].

Hansen, H., (1980) Reconstruction of the electron momentum distribution from a set of experimental Compton profiles, Hahn Meitner Institute (Berlin), Report HMI B 342. 

Cisneros GA, Piquemal J-P, Darden TA (2006) Generalization of the Gaussian electrostatic model extension to arbitrary angular momentum, distributed multipoles and speedup with reciprocal space methods. J Chem Phys 125 184101... 

Fig. 9. (a) Momentum distributions of CD3 and C5D3. Momentum of these two frag-... 

Fig. 24. Ion image of photofragment (a) m/e = 98, (b) m/e = 18 from photodissociation of d. .-cl liyIboii/ono at 193 nm. The delay times between pump and probe laser pulses are 30 /is and 7 //s. respectively, (c) The translational momentum distributions of m/e = 18 (thin solid line) and 98 (thick solid line), (d) The fragment translational energy distribution for the reaction C6D5C2D5 —> C6D5CD2 + CD3.

The analysis is performed for the calculations with rrot=0 K for the CH3C1 reactant, so that the angular momentum distribution for the complex P(j) is the distribution of orbital angular momentum for complex formation P(i). This latter distribution is given in ref. 37. Jm , the quantum number for j, varies from 282 for Enl = 0.5 kcal/mol to 357 for rel = 3.0 kcal/mol. The term k iEJ) in equation 24 is written as k (.EJ)=k Ejyf E), where k EJ) is the classical RRKM rate constant with the CH3C1 intramolecular modes inactive and / ( ) is treated as a fitting factor. 

A two-dimensional example of a spherical and a skewed momentum distribution is shown in Fig. 8.2 for this simple case, es = (J), and the variance along es has... 

The objective of the method presented here is to develop a momentum distribution that will bias path dynamics along the slow manifold, permitting the efficient calculation of kinetic properties of infrequent reactions. 

A detailed numerical implementation of this method is discussed in [106]. W is the statistical weight of a trajectory, and the averages are taken over the ensemble of trajectories. In the unbiased case, W = exp -(3Wt), while in the biased case an additional factor must be included to account for the skewed momentum distribution W = exp(-/ Wt)w(p). Such simulations can be shown to increase accuracy in the reconstruction using the skewed momenta method because of the increase in the likelihood of generating low work values. For such reconstructions and other applications, e.g., to estimate free energy barriers and rate constants, we refer the reader to [117]. 

By integrating out the coordinate dependence, they also obtain an approximate quantum-corrected formula for the momentum distribution which leads to the definition of an effective temperature. That is, the approximate distribution is still Gaussian in the momenta, but with an increased temperature for each particle... 

---