State sum

Lin S H and Villaeys A A 1994 Theoretical description of steady-state sum-frequency generation in molecular absorbates Phys. Rev. A 50 5134-44... 

Calculate activation energies for the three Diels-Alder reactions (energy of transition state - sum of energies of reactants). Which reaction has the smallest energy barrier Which has the largest energy barrier Do your results parallel the measured relative rates of the same reactions (see table at left) ... 

An explanation for the isotope effects was given in terms of differences in the zero-point energies of the transition states and the influence of slight reductions of isotope-dependent frequencies on the state sums. 

Here it is now understood that Vh = fen + fes and are no longer to be calculated from the approximate TFD Euler equation (4) but rather using a density n(r) given by the ground-state sum... 

F(J ) is the pair interaction potential, T the temperature and k the Boltzmann factor. The integral to the right is over the volume v of the container it is often referred to as the configuration integral, Q2(T). For large enough volumes, the integral may be replaced by the volume. With the help of Hill s effective potentials, Eq. 2.28, this expression may be written as a sum of free and bound pair state sums, respectively [183, 184],... 

The process has been treated theoretically in terms of simplified models.14 58 The quantum mechanics is one of formulating the probability of crossing from an excited to a ground state, summed over all vibrational levels. For coordination compounds, the weak coupling limit is presumably the important approximation. Here, the transition is from low lying vibrational levels of the excited state to very high vibrational levels of the ground state. 

The transition state sum omits the reaction coordinate degree of freedom since it is not a bound vibration and does not contribute to the zero-point energy in the transition state. E% and Er are respectively the energy of the potential energy surface at transition state and reactants. Then,... 

Fig. 3. Comparison of classical and semi-classical state sums for a single oscillator.

We should mention that another possible explanation of the Pauli-like behavior has recently been proposed in terms of contributions of the triplet-excited bipolaronic states to the spin susceptibility [94]. As shown by Bussac and Zuppiroli, the polaron-bipolaron energy difference U is essentially determined by the interdopant distance Ld. The authors also consider the bipolaron triplet state, which reduces to two separate polarons for large Ld. Due to disorder, the Ld are distributed, giving rise to a distribution for the energy of the magnetic states. Summing over the distribution yields a Curie-like contribution, plus a smoothy temperature-dependent term, which resembles a Pauli contribution. 

Z force component in z-direction state sum ( partition func-... 

In contrast with first derivatives, second derivatives involve couplings with all states (sums over M in (A.15)) that correspond to a second-order Jahn-Teller effect. Such contributions from higher-lying states (M > 2) do not exist in a pure two-level model (see Sec. 2), but they are part of the actual MCSCF calculation, where the number of eigenstates is equal to the number of CSFs. Limiting the values of M to 0 and 1 leads to ... 

The intrinsic state of an atom can be simply thought of as the ratio of it and lone-pair electrons over the count of the g bonds in the molecular graph for the atom considered. Therefore, the intrinsic state reflects the possible partitioning of non-0 electron s influence along the paths starting from the considered atom the less partitioning of the electron influence, the more available are the valence electrons for intermolecular interactions. The sum of the intrinsic states of all of the atoms is a molecular descriptor called intrinsic state sum moreover, from the intrinsic state sum the —> Q polarity index was derived. 

H3 represents an oxidation-state sum of +3( 4-1 for each of the three hydrogens). Then the oxidation state of P must be -3, since the algebraic sum of the oxidation states of all atoms in a compound must equal zero. 

This expression looks just like the continuous-wave differential cross-section for a fixed nuclear geometry in a single electronic state (1), weighted by the probability distribution of nuclear geometries in the populated electronic states, summed up over the duration of the incoming X-ray pulse and, finally, weighted by the... 

Here N is the total number density of molecules, Qj and Qv are the rotational and vibrational state sums, and gj=2 / +1 is the statistical weight of the level. Qj and Qv are approximated as... 

The accurate cumulative reaction probability and density of reactive states (summed over parities) for 7 = 4 are shown in Fig. 3. The nine prominent features correspond to transition states with v2 = 0, 1, or 2 for v, = 0, 1, and 2, just as in the 7 = 1 spectrum (9). States with v2 > 2 also occur, but they are harder to identify because they are broad and... 

Feature Energy (eV) States Sum of degeneracies of these states Running sum N ... 

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