Expanding the exponent

This result is always attractive. The reason is again very simple. At any orientation for which f/(Qj, positive, U>0, there exists another orientation (obtained by inverting the direction of one dipole) for which U<0. The weights given by the Boltzmann factors are now different, we have exp(- 3L0 > 1 for C/ < 0, and exp(-piT) < 1 for f/ > 0. The average is therefore biased in favor of the attractive orientations, therefore the net result is < 0. We note also that when either the distance or the temperature increases so that ipt/1 < < 1, we can expand the exponent in Eq. (1.3.7) to obtain... 

Expand the exponent in Eq. (4.17) into power series, we obtain... 

Strict e-expansion cannot yield this result. It expands the exponent according to 1 jv 2—g/4-h0(e2) (cf. Eq. (8,34)) and yields a power series in ln(q2i 2) ... 

We can do some general conclusions, based on the form of the tunneling Hamiltonian (162). Expanding the exponent in the same way as before, we... 

Now we can expand the exponent (note that S-operator is defined only in the sense of this expansion)... 

However, if Ed ErD, the adiabatic approach proves valid even at a sufficiently large distance r. This inequality permits us to expand the exponent in formula (56) on the small value ErDj ED. Then, the exponent... 

The physical sense of the condition of the violation of the adiabatic approach in the representation of the wave functions can be easily understood if we expand the exponent in the expression (64) on e and turn to the usual units ... 

The F-K s method of expanding the exponent is applied to make the exponential term integrable. Another example is presented with regard to Eqs. (25) (39) in Section 2.4. For this method, refer to a footnote in Section 2.7 as well. 

When the temperature, T, of 2 cm of a chemical of the TD type, irrespective of liquid and powdery, including every gas-permeable oxidatively-heating substance, charged, or confined, in some one of the open-cup, the draft or the closed cell, in accordance with the self-heating property of the chemical, and subjected to either of the two kinds of adiabatic tests started each from a T is immediately above the T, if two conditions, i.e., (T - r,)<Frank-Kamenetskii s method of expanding the exponent, i.e., Eqs. (25) and (26) presented in Section 1.3,... 

After having been converted into an integrable form by applying the method of expanding the exponent, i.e., Eqs. (25) and (26) presented in Section 1.3 [8], Eq. (53) is integrated to yield Eq. (54). 

In this regard, it may be permitted to say that the Frank-Kamenetskii s method of expanding the exponent, which has been applied in Sections 1.3 and 2.4, is, in the result, nothing else but the linear approximation of the exponential function. 

Again, expanding the exponent in (7.125) in a power series and comparing similar orders of x(Z) in the resulting series with Eq. (7.124) we find... 

If we expand the exponent in powers of the quantity on the right, we obtain finally the approximation... 

In case of the potential (61) we can expand the exponent in/ and in the case of a short ranged potential we can as already mentioned change to ZZfoJ i.e., just cancel / / ... 

Suppose that the temperature is sufficiently high, so that the condition tim < kT holds. Then one can neglect the second term in the right-hand side of (7.16) and expand the exponent term into a series retaining only first two terms. As a result, the formula takes the form Eia = kT. Hence, under this assumption we obtain... 

Another structural aspect of the CCSD wave function is related to its exponential form (3.1). The cumulative value of the configuration coefficient of a particular determinant is determined by expanding the exponent in a Taylor serious in terms of the singly and doubly excited CC operators and collecting terms generating this particular determinant. The configuration coefficient of a particular determinant (say, determinant ((A ), where k indicates the level of excitation of the determinant with respect to the reference determinant and is the determinant number in the manifold of the k excited determinants) can be determined as ... 

Such a form of quasi-equilibrium distribution takes place due to the fact of the availability of two invariants of motion. In Equation 25 parameters a and p linked to the operators Hz and Hss are thermodynamically conjugative parameters for the Zeeman energy and the energy of spin-spin interactions respectively. We can expand the exponent in Equation 25 in jxjwers of xT-Lz and f Hss and keep only the linear terms. As we shall see later such a linearization corresponds to the high temperature approximation. In the linear approximation in x Hz and Hss, the density matrix is reduced to... 

Distribution function 187, 196 describes random walking of segments in the external potential field , so G(/Vo,4>) is written as a series in on expanding the exponent in Equation 196 in (Freed, 1972) (cf. Equation 24)... 

After expanding the exponents in Eq. 2.42 in a Taylor series and only maintaining the first two terms, the energy eigenvalues are inserted to arrive at... 

Another interpretation of may be given as follows. For xcr 1, we may expand the exponent to the first order to obtain... 

In the thormgtl regime the corresponding expressions can be obtained by expanding the exponent in (2.1.2). Replacing the sum over m by the integration over the first Brillouin zone... 

The cumulants method is a moment expansion method. It compresses the entire multi-exponential decay distribution into the exponent and then expands the exponent F to a polynomial expression of the cumulants (the moments) and is ... 

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