Spin partitioning functional

The one-spin partition function = J- j dz exp(CTz ) can be written in terms of error functions of real and imaginary argument as... 

The partition function for a molecule is formed of the partition functions for individual types of energy increments (motions), i.e. from the translational, rotational, internal rotational (free rotation, hindered rotation), vibrational, electronic and nuclear spin partition functions... 

The nuclear spin partition function is given by (2i + 1) where i is the nuclear spin quantum number, since the energy of nuclear orientations is very small compared with kT Thus, for the hydrogen atom ( H), / = 1/2 and for the chlorine atom i = 3/2 giving nuclear spin contributions of 2 and 4, respectively, to the partition function. However, it is conventional to omit these factors from calculated entropies. 

Nuclear Spin Contribution. The nuclear spin partition function is the product of the nuclear spin multiplicity 2ij -H 1) for all the atoms in the molecule, where /V is the nuclear spin of the yth atom. Since, apart from processes involving molecular hydrogen and its isotopes (and the transmutation of the elements), nuclear spins are conserved, this contribution is conventionally omitted from the total entropy leaving the practical or virtual entropy (these adjectives are frequently omitted also). 

Calculate the equilibrium constant K for temperature T = 5000 K. There is no rotational or vibrational partition function for H, H+, or e , but there are spin partition functions Qs = 1 forH+, and qs = 2 fore . AD = -311kcalmol... 

K i) and i a( a)>0 and are Lagrange multipliers enforcing constraints (5.21) and (5.22) co is the unit sphere surface comprised of all possible endpoints for h, and integration over m means integration over all possible choices for h This spin partitioning functional has a unique minimum that is found by an iterative solution algorithm. ... 

There are in principle also energy levels associated with nuclear spins. In the absence of an external magnetic field, these are degenerate and consequently contribute a constant term to the partition function. As nuclear spins do not change during chemical reactions, we will ignore this contribution. 

Notice that the associated spin model has the following three properties (1) it is, in general, anisotropic (i.e. a-2 / CI3), (2) its set of coupling constants hi, hij, /1123) are interdependent (this should be obvious from equation 7.63, which provides a parameterization of each of these seven constants in terms of our original four independent conditional probabilities, aj (equation 7.58)), and (3) its partition function, Z, can be calculated exactly. 

Particles spin Vz, 517 Dirac equation, 517 spin 1, mass 0,547 spin zero, 498 Partition function, 471 grand, 476 Parzen, E., 119,168 Pauli spin matrices, 730 PavM, W., 520,539,562,664 Payoff, 308 function, 309 discontinuous, 310 matrix, 309... 

In Eq. (44), gei(T ) is the ratio of transition state and reactant electronic partition functions [31] and the rotational degeneracy factor = (2ji + l)(2/2 + 1) for heteronuclear diatomics, and will also include nuclear spin considerations in the case of homonuclear diatomics. 

The first illustration of the concept of a partition function is that of a two-level system, e.g. an electron in a magnetic field, with its spin either up or down (parallel or anti parallel to the magnetic field) (Fig. 3.2). The ground state has energy Eq = 0 and the excited state has energy Ae. By substituting these values in Eq. (3) we find the following partition function for this two-level system ... 

If only the spins of the nuclei are altered in a given transition, the translational and vibrational contributions to the partition function are identical. Thus, for tiie reaction ortho-Hz - para-H2, the partial-pressure ratio at equilibrium is given by... 

The remark just made suggests that a natural place to begin our discussion of equilibrium equations is with the occupation of different charge states. Let a hydrogen in charge state i(i = +, 0, or - ) have possible minimum-energy positions in each unit cell, of volume O0, of the silicon lattice. (O0 contains two Si atoms, so our equations below will be applicable also to zincblende-type semiconductors.) To account for spin degener-ancies, vibrational excitations, etc., let us define the partition function... 

Proceeding in the spirit above it seems reasonable to inquire why s is equal to the number of equivalent rotations, rather than to the total number of symmetry operations for the molecule of interest. Rotational partition functions of the diatomic molecule were discussed immediately above. It was pointed out that symmetry requirements mandate that homonuclear diatomics occupy rotational states with either even or odd values of the rotational quantum number J depending on the nuclear spin quantum number I. Heteronuclear diatomics populate both even and odd J states. Similar behaviors are expected for polyatomic molecules but the analysis of polyatomic rotational wave functions is far more complex than it is for diatomics. Moreover the spacing between polyatomic rotational energy levels is small compared to kT and classical analysis is appropriate. These factors appreciated there is little motivation to study the quantum rules applying to individual rotational states of polyatomic molecules. 

Recently, the concept of thermal entanglement was introduced and studied within one-dimensional spin systems [64-66]. The state of the system described by the Hamiltonian H at thermal equilibrium is p T) = exp —H/kT)/Z, where Z = Tr[exp(—7//feT)] is the partition function and k is Boltzmann s constant. As p T) represents a thermal state, the entanglement in the state is called the thermal entanglement [64]. 

We are now done with spin functions. They have done their job to select the correct irreducible representation to use for the spatial part of the wave function. Since we no longer need spin, it is safe to suppress the subscript in Eq. (5.110) and all of the succeeding work. We also note that the partition of the spatial function X is conjugate to the spin partition, i.e., 2"/ , 2. From now on, if we have occasion to refer to this partition in general by symbol, we will drop the tilde and represent it with a bare X. 

In particular instances, the above approximation is insufficiently accurate because one or more excited electronic states lie close in energy to the ground state. A typical example occurs for heavy halogen atoms, where spin-orbit coupling creates 2P 2 and states having only a narrow energy separation. In such cases, explicit formation of the partition function cannot be avoided, but even then one typically need only include a small number of terms in the total sum, and evaluation is not unduly taxing. 

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