Lowest energies

Measurement of the energy difference is achieved by a resonance method. The population of nuclei in a given state is governed by the Boltzman distribution that leg s to an of nuclei in the state of lowest energy and... 

To illustrate the use of the variational prineiple, results are presented here for ealeulations of the five lowest energy states (the ground state and the first four exeited states) of a partiele subjeet to the potential... 

The orthogonality condition assures one that the lowest energy state will not converge to core-like states, but valence states. The wavefimction for the solid can be written as... 

Many substances exist in two or more solid allotropic fomis. At 0 K, the themiodynamically stable fomi is of course the one of lowest energy, but in many cases it is possible to make themiodynamic measurements on another (metastable) fomi down to very low temperatures. Using the measured entropy of transition at equilibrium, the measured heat capacities of both fomis and equation (A2.1.73) to extrapolate to 0 K, one can obtain the entropy of transition at 0 K. Within experimental... 

In an ideal Bose gas, at a certain transition temperature a remarkable effect occurs a macroscopic fraction of the total number of particles condenses into the lowest-energy single-particle state. This effect, which occurs when the Bose particles have non-zero mass, is called Bose-Einstein condensation, and the key to its understanding is the chemical potential. For an ideal gas of photons or phonons, which have zero mass, this effect does not occur. This is because their total number is arbitrary and the chemical potential is effectively zero for tire photon or phonon gas. 

Figure Bl.4.8. Top the lowest-energy structures of water clusters, from n = 2 -6. Bottom a sample...

Methods that are based on making the fiinctional (T // T ) / ( T T ) stationary yield upper bounds to the lowest energy state having the synnnetry of the CSFs in T. The Cl and MCSCF methods are of this type. 

QMC teclmiques provide highly accurate calculations of many-electron systems. In variational QMC (VMC) [112, 113 and 114], the total energy of the many-electron system is calculated as the expectation value of the Hamiltonian. Parameters in a trial wavefiinction are optimized so as to find the lowest-energy state (modem... 

The discussion in the previous sections assumed that the electron dynamics is adiabatic, i.e. the electronic wavefiinction follows the nuclear dynamics and at every nuclear configuration only the lowest energy (or more generally, for excited states, a single) electronic wavefiinction is relevant. This is the Bom-Oppenlieimer approxunation which allows the separation of nuclear and electronic coordinates in the Schrodinger equation. 

Koga N and Morokuma K 1985 Determination of the lowest energy point on the crossing seam between two potential surfaces using the energy gradient Chem. Phys. Lett. 119 371... 

After this, Martinez and Ben-Nun applied the method to the photoexcitation of ethylene [88,247]. The lowest energy excitation is the HOMO-LUMO n n transition. These states are labeled A and Close in energy to... 

A simple VB approach was used in [75] to describe the five structures. Only the lowest energy spin-pairing structures I (B symmehy) of the type (12,34,5 were used (Fig. 21). We consider them as reactant-product pairs and note that the transformation of one structure (e.g., la) to another (e.g., Ib) is a thr ee-electron phase-inverting reaction, with a type-II transition state. As shown in Figure 22, a type-II structure is constructed by an out-of-phase combination of... 

Such renormalization can be obtained in the framework of the small polaron theory [3]. Scoq is the energy gain of exciton localization. Let us note that the condition (20) and, therefore, Eq.(26) is correct for S 5/wo and arbitrary B/ujq for the lowest energy of the exciton polaron. So Eq.(26) can be used to evaluate the energy of a self-trapped exciton when the energy of the vibrational or lattice relaxation is much larger then the exciton bandwidth. 

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