Molecular partition function quantum

The second product is over the 3N—6(3N—5) normal mode frequencies of the ideal gas harmonic molecule to which Equation 4.78 applies. Thus the product over vibrations Equation 4.90 is indeed the quantum mechanical contribution to the molecular partition function for the ideal gas. 

This result indicates that (s2/si)f (compare Equations4.78 and 4.93) is just the quantum effect on the molecular partition functions of the normal mode vibrations. This result has now been derived without the explicit use of the Teller-Redlich product rule. 

The information needed to evaluate the molecular partition functions qb (13.63), may in principle be obtained from experimental spectroscopic measurements or theoretical calculations on each molecule i. Each type of energy contribution to qt (translational, rotational, vibrational, electronic) in principle requires associated quantum energy levels... 

The sum in Eq. (29.30) is over all the quantum states of the molecule, so q is the molecular partition function. If quantum states have the same energy, they are said to be degenerate degeneracy = gi. The terms in the partition function can be grouped according to the energy level. The states yield g equal terms in the partition function. The expression in Eq. (29.30) can be written as... 

Equation (29.75) is an important link between quantum mechanics and chemistry. Knowing the energy levels of the molecules, we can calculate the molecular partition functions. Then we use Eq. (29.75) to obtain the equilibrium constant for the chemical reaction. 

The pyrolysis mechanism of PPE and its derivatives given in Scheme 7.2 consist of bimolecular and unimolecular reactions. Applying transition state theory, we calculate the rate constants for the hydrogen abstraction reactions using Eq. (7.19) and the rate constants for reactions 1 and 3-5 using Eq. (7.21). The Wigner correction (Eq. (7.20)) is utilized to approximate quantum effects and the molecular partition functions are defined through Eqs (7.14), (7.15), (7.17), (7.18), and (7.28). 

The molecular partition function may thus be interpreted as equal to N/tiQ, the ratio of the total number of molecules to the number in the lowest quantum state. Equation (12 72) can similarly be used to give a meaning to each term which occurs in / for example, the term is proportional to the mean population of the particular level Thus each term in the molecular partition function is proportional to the... 

Quantum mechanics helps determine the energies of molecules and the molecular partition functions, as we will see in Chapter 6, can be calculated from the characteristics of molecules (mass, moments of inertia, vibrational frequencies, etc.). The aim of this section is to determine the value of the canonical partition functions, since this is useful to link them to molecular partition functions. 

The translational factor of the classical molecular partition function is proportional to the quantum mechanical translational factor ... 

Equation 11.30 gives a sum over the quantum energy levels of a molecular system, the molecular partition function, when the molecular energies (Equation 11.1) are used for the e/s. (The parallel definition for the partition functions of systems of particles uses the... 

Thermodynamic properties of an assembly of many mutually independent molecules of the same kind are determined by one single quantity, the so called molecular partition function. The partition function is a statistical mechanics concept representing a link between microscopic and macroscopic thermodynamic properties. It enables the expression of the equilibrium constant in terms of energies of individual degrees of freedom for the molecules. The molecular partition function is defined as the sum over stationary quantum states of the molecules (numerated by i = 1,2,3,...) ... 

Next, Ah and Ad are written in terms of partition functions (see Section 5.2), which are in principle calculable from quantum mechanical results together with experimental vibrational frequencies. The application of this approach to mechanistic problems involves postulating alternative models of the transition state, estimating the appropriate molecular properties of the hypothetical transition state species, and calculating the corresponding k lko values for comparison with experiment.""- " "P... 

Approximate calculations of this activation energy have been made in a number of examples using the quantum theory of molecular binding, by making assumptions concerning the structure and partition functions of the transition state molecule. 

Fig. 6.1 An aspartate amino acid partitioned into quantum and classical (MM) regions. The functional group of the side chain, involved in the chemical reaction, lies within the quantum region and the backbone atoms are treated by using a molecular mechanics force field.

The result (Equation 4.90) could have been derived more simply. It has been emphasized that the quantum mechanical contribution to the partition function ratio arises from the quantization of vibrational energy levels. For the molecular translations and rotations quantization has been ignored because the spacing of translational and rotational energy levels is so close as to be essentially continuous (As/kT 1). 

Schrodinger equation. When the molecule is too large and difficult for quantum mechanical calculations, or the molecule interacts with many other molecules or an external field, we turn to the methods of molecular mechanics with empirical force fields. We compute and obtain numerical values of the partition functions, instead of precise formulas. The computation of thermodynamic properties proceeds by using a number of techniques, of which the most prominent are the molecular dynamics and the Monte Carlo methods. 

Following Fey nman s original work, several authors pmsued extensions of the effective potential idea to construct variational approximations for the quantum partition function (see, e g., Refs. 7,8). The importance of the path centroid variable in quantum activated rate processes was also explored and revealed, which gave rise to path integral quantum transition state theory and even more general approaches. The Centroid Molecular Dynamics (CMD) method for quantum dynamics simulation was also formulated. In the CMD method, the position centroid evolves classically on the efiective centroid potential. Various analysis and numerical tests for realistic systems have shown that CMD captures the main quantum effects for several processes in condensed matter such as transport phenomena. 

For the vibrational term qivib, a classical high-T continuum approximation is seldom valid, and evaluation of the discrete sum over states is therefore required over the quantum vibrational distribution. (As pointed out in Sidebar 5.13, accurate treatment of molecular vibrations is crucial for accurate assessment of entropic contributions to AGrxn.) A simple quantum mechanical model of molecular vibrations is provided by the harmonic oscillator approximation for each of the 3N — 6 normal modes of vibration of a nonlinear polyatomic molecule of N atoms (cf. Sidebar 3.8). In this case, the quantum partition function can be evaluated analytically as... 

In addition to being a function of T, the partition function is also a function of V, on which the quantum description of matter tells us that the molecular energy levels, , depend. Because, for single-component systems, all intensive state variables can be written as functions of two state variables, we can think of q(T, V) as a state function of the system. The partition function can be used as one of the independent variables to describe a single-component system, and with one other state function, such as T, it will completely define the system. All other properties of the system (in particular, the thermodynamic functions U, H, S, A, and G) can then be obtained from q and one other state function. 

The above formula for Z, the NPT partition function, was first reported by Guggenheim [74], who wrote the expression down by analogy rather than based on a detailed derivation. While this form of the partition function is thought to be broadly valid and is widely applied (for example in molecular dynamics simulation [6]), it introduces the conceptual difficulty that the meaning of the discrete volumes Vi is not clear. Discrete energy states arise naturally from quantum statistics. Yet it is not necessarily obvious what discrete volumes to sum over in Equation (12.50). In fact for most applications it makes sense to replace the discrete sum with a continuous volume integral. Yet doing so results in a partition function that has units of volume, which is inappropriate for a partition function that formally should be unitless. 

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