The high temperature approximation

Of the 3N normal modes of vibration of the crystal, those with the lowest frequencies are the ordinary acoustic vibrations which appear in the theory of soimd. On the other hand, the highest possible frequency is determined by the smallest possible independent wave-len h, and this can be shownf to be of the order of cm, the 

Consider the largest possible value of hvjkT. Using the known values of h and Ir, this ratio may be calculated to be about 60/T or QOOITy according to whether the largest frequency of the crystal is of the order 10 or 10 respectively. Clearly this ratio will be small compared to unity at sufficiently high temperatures and for certain crystals, those for which the highest frequency is about 10, even at room temperature. 

Under conditions of temperature where hvlkT ly the terms beyond the third in the expansion 

It may be remarked that transition metals have values of Cy which considerably exceed 3R at high temperatures for example, y iron has a value of about 38 JK mol . This is believed to be due to electronic excitation from the unfilled d shells, a factor which is not allowed for in the theory as developed above. The applicability of the Dulong and Petit r e to ionic and molecular lattices will be discussed in a later section. 

The same result as (13 45) could also be obtained by applying the principle of equipartition ( 12 12) which will be valid under conditions where hvjkT i. Each oscillator will have a mean energy of kT, as shown in equation (12 111), and the total thermal energy of the crystal is therefore SNk T in agreement with (13 45). 

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In the presence of a potential function U(x,y), the density matrix in the high-temperature approximation has the fomi... 

The experiment starts at equilibrium. In the high-temperature approximation, the equilibrium density operator is proportional to the sum of the operators, which will be called F. If there are multiple exchanging sites with unequal populations, p-, the sum is a weighted one, as in equation (B2.4.31). 

This universality is peculiar for the high-temperature approximation, which is valid for //J < 1 only. For sufficiently high temperature the quantum theory confirms the classical Langevin theory result of J-diffusion, also giving xj = 2xE (see Chapter 1). This relation results from the assumed non-adiabaticity of collisions and small change of rotational energy in each of them ... 

NFS spectra of the molecular glass former ferrocene/dibutylphthalate (FC/DBP) recorded at 170 and 202 K are shown in Fig. 9.12a [31]. It is clear that the pattern of the dynamical beats changes drastically within this relatively narrow temperature range. The analysis of these and other NFS spectra between 100 and 200 K provides/factors, the temperature dependence of which is shown in Fig. 9.12b [31]. Up to about 150 K,/(T) follows the high-temperature approximation of the Debye model (straight line within the log scale in Fig. 9.12b), yielding a Debye tempera-ture 6x) = 41 K. For higher temperatures, a square-root term / v/(r, - T)/T,... 

Using the wave functions of the harmonic oscillator in each potential well of the proton, we can estimate the total effect of the inertia on the transition probability in the high-temperature approximation for the medium67 ... 

A spin system, initially in thermal equilibrium, can be described by a density operator high temperature approximation.3 After a 90° pulse, the system will evolve under the Hamiltonian and the density operator cr(t) in the rotating frame with a speed a)tf=2nftf can be written as3,4... 

The second group contains the low frequencies, m < 1, which are to be treated in the high temperature approximation. This, we have seen, accounts for excitation into upper levels by expanding I n(s/s ) 1 in even powers of u (see Chapter 4)... 

We start at equilibrium. In the high-temperature approximation, the equilibrium density operator is proportional to the weighted sum of the operators, which we will call 4- We assume that a simple, non-selective pulse has been used at the start of the experiment. This rotates the equilibrium 2 magnetization onto the x axis. After the pulse the density matrix is therefore given by 4, and it will evolve as in equation (7) or (8). If we substitute (8) into (10), we get the NMR signal as a function of time t, as given by (11). The detector sees each spin (but not each coherence ) equally well. 

Further Mossbauer effect studies (304 — 12 K) and magnetic susceptibility measurements (301 - 1K) on the neutral complex [Fe(papt)2 ] have been performed recently 1S9 The magnetic data are shown in Fig. 34. The values of - In f(s T2) and - In f(J Aj) have been found to follow the high-temperature approximation of the Debye model above 105 K and 140 K, respectively, if anharmonic corrections have been introduced. No simple model is available at present, which would be capable to account for the complete temperature dependence of the Debye-Waller factors in this crossover system. 

Even for the largest magnetic fields available for NMR, the energy levels are separated only by mi/Iicalories, and the argument in the exponential is very small except at extremely low temperature. Hence, the high temperature approximation e x 555 1 — x may be employed to show that the fractional excess population in the lower level is... 

On is the vibrational Raman transition frequency, and Yn is the vibrational dephasing constant Within the high temperature approximation, Eq.(22) is expressed as... 

In the physical picture ion-pairs are just consequences of large values of the Mayer /-functions that describe the ion distribution [22], The technical consequence, however, is a major complication of the theory the high-temperature approximations of the /-functions applied, e.g. in the mean spherical approximation (MSA) or the Percus-Yevick approximation (PY) [25], suffice in simple fluids but not in ionic systems. 

A solution to these difficulties is a blend of the chemical picture in which clustered ion configurations are described by the mass action law, while the interactions between the various entities are treated by methods applying the high-temperature approximations of the /-functions, e.g. by the MSA. The Debye-Hiickel (DH) theory [26], although derived from classical electrostatics, is also a high-temperature approximation, whose range of applicability can be extended by supplementing a mass action law for ion pair formation [27],... 

Transport Coefficients of Qnantum-Classical Systems 543 where we used the high temperature approximation... 

These effects on NMR can either be described by a classical treatment of the demagnetization field or by a macroscopic quantum treatment which removes several important simplifications such as the concept of the molecular spin system and the rejection of the high-temperature approximation. However, Levitt has noted that only the classical treatment produces quantitative agreement with experiment. 

Equation 68 is referred to as the high-temperature approximation because the potential appears in the integral as the product 0 V. Thus, the approximation is valid for small perturbations or high temperatures. Higher-order terms can be included, in principle, to obtain more accuracy. However, the derivatives of the distribution functions with respect to X involve higher-order distribution functions 170 e.g., the first-order correction in X to the distribution function involves three- and four-body distribution functions which are usually difficult to obtain. In some cases, the superposition approximation or other approximate expressions for the higher-order distributions have been introduced.175 However, the first-order result is the one that has been employed in most applications.176... 

The initial density matrix at the moment when f = 0, i.e. before the pulse excitation, in the high-temperature approximation can be written as... 

The high-temperature approximation is certainly valid here. 

In conclusion, we would like to emphasize again that in the general case of a non-separable reaction coordinate, the high temperature approximation to the classical (semiclassical) collision theory. 

The same interpretation results from the above approximate treatment of radical recombination reactions, using the "diatomic" model, where the collision diameter d is related to the high temperature approximation of expression OOliV) for the rotational partition function of product molecule AB, being in a state which should be considered a.non-stationary transition state. 

D17.1 An approximation involved in the derivation of all of these expressions is the assumption that the contributions from the different modes of motion are separable. The expression = kT/hcB is the high temperature approximation to the rotational partition function for nonsymmetrical linear rotors. The expression q = kT/hcv is the high temperature form of the partition function for one vibrational mode of the molecule in the haimonic approximation. The expression (f- =g for the electronic partition function applies at normal temperatures to atoms and molecules with no low lying excited electronic energy levels. 

Also sio is the si function, defined by Eq. (12), for the reference system. This is sometimes called the high-temperature approximation, because it represents the first two terms of a series for si in powers of i/kT. While it might be... 

The high-temperature approximations are also systematically more accurate with the truncation parameter L, but being truncated Taylor series, they tend to become unstable abruptly at one point. The imaginary-time splitting scheme renders stability to this approximation and gives accurate estimates of U in the wide range of temperature from 10 K to 10 K with just... 

Fig. 2 The entropy (S) of hydrogen fluoride calculated with grand canonical FCI/STO-3G and its various approximations including MP0/STO-3G as a function of temperature (7). M is the number of states (in each group of determinants with the same number of electrons and magnetic spin quantum number) included in the low-temperature approximation. L is the number of terms in the Taylor series expansion in the high-temperature approximation, where the imaginary-time splitting is not used. The plot of S from canonical FCI/STO-3G is also superimposed...

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