Field theories

Theory Bethe. As originally conceived, it was a model based on a purely electrostatic 

The five d orbitals in an isolated, gaseous metal ion are degenerate. If a spherically symmetric field of negative charges is placed around the metal, the orbitals will remain degenerate, but all of them will be raised in energy as a result of the repulsion between 

14 The evolution of bonding models for inorganic complexes from the 1920s to the mid-1950s is described in Balhausen. C. J. J. Client. Eiluc. 1979.16. 194-197. 215-218. 357-361. 

To gain some appreciation for the magnitude of A and how it may be measured, let us consider the cf1 complex, [Ti(H30)sI3, . This ion exists in aqueous solutions of Tt3+ and gives rise to a purple color. The single d electron in the complex will occupy 

The absorption spectrum of [Ti(H,0)6]3+ (Fig. 11.8) reveals that this transition occurs with a maximum at 20,300 cm-1, which corresponds to 243 kJ mol-1 of energy for A .17 By comparison, the absorption maximum 8 of ReFfi (also a d1 species) is 32,300 cm-1, or 338 kJ mol 1. These are typical values for A and are of the same order of magnitude as the energy of a chemical bond. 

The model that largely replaced valence bond theory for interpreting the chemistry of coordination compounds was the crystal field theory, first proposed in 1929 by Hans Bethe. As originally conceived, it was a model based on a purely electrostatic 

7 Splitting of the five d orbitals by an octahedral field. The condition represented by the degenerate levels on the len is a hypothciical spherical field. 

An interesting and useful method of theoretical treatment of certain properties of complexes and crystals, called the ligand field theory, has been applied with considerable success to octahedral complexes, especially in the discussion of their absorption spectra involving electronic transitions.66 The theory consists in the approximate solution of the Schrddinger wave equation for one electron in the electric field of an atom plus a perturbing electric field, due to the ligands, with the symmetry of the complex or of the position in the crystal of the atom under consideration. 

In some respects the ligand field theory is closely related, at least qualitatively, to the valence-bond theory described in the preceding sections, and many arguments about the structure of the normal state of a complex or crystal can be carried out in either of the two ways, with essentially the same results.66 

For example, it has been found66 that CrF crystallizes with the rutile structure (Fig. 3-2), but with four of the Cr—F bonds (lying in a plane) with length 2.00 0.02 A and the other two with length 2.43 A (and hence presumably much weaker), whereas in other crystals (MgFj, TiOi) the six ligands of the metal atom are at essentially the same distance. The distortion of the coordination polyhedron can be ex- 

CrF forms a cubic crystal containing regular octahedral CrFe groups, each fluorine atom forming a joint corner of two octahedra the Cr—F bonds all have length 1.90 A. The regularity of these octahedra is expected the three unpaired 3d electrons use only three of the 3d orbitals, leaving two available for formation of d sjfi octahedral bonds. 

In an environment with regular octahedral symmetry the five d orbitals can be divided into two sets. Two orbitals, d,a and inter- 

In 1951, chemists trying to make sense of metal complex optical spectra and color returned to an emphasis on the ionic nature of the coordinate covalent bond. Coordination chemists rediscovered physicists Hans Bethe s and John van Vleck s crystal field theory (CFT), 

For a d4 t2g3eg1 configuration in an octahedral field for example, the CFSE is  

One of the factors governing the magnitude of Ac is the nature of the bound ligands. For a metal ion, the field strength increases according to the spectrochemical series  

We have held this discussion as general and qualitative as possible, in order not to get into the (obviously debatable) details of each model (4). It is characteristic for all Huckel-Wolfsberg-Helmholz types of models that the delocalization of the anti-bonding orbital becomes more pronounced when the denominator (oim — ax) decreases. In the case of m = ax one would replace Eq. (27) by 

This procedure has thus far been applied to terminally blocked alanine (with (j , t i, and x as variables) and to Met-enkephalin (with j , and 4 , of each residue as variables, but with the to/s and side-chain dihedral angles fixed at the values corresponding to the known global minimum). The global minimum was achieved in several trial runs for each of these molecules. 

CHAPTER 22 The Transition Elements and Their Coordination Compounds 

Electrons in the d orbitals of the free metal ion experience an average net repulsion in the negative ligand field that increases all d-orbital energies. Electrons in the fjg set are repelled less than those in the Sg set. The energy difference between these two sets is the crystal field splitting energy, A. 

Average potential energy of 3d orbitals raised in octahedral ligand field 

The substance has a color because only certain wavelengths of the incoming white light are absorbed. 

Consider the [Ti(H20)6] ion, which appears purple in aqueous solution 

Absorption spectra show the wavelengths absorbed by a given metal ion with different ligands and by different metal ions with the same ligand. From such data, we relate the energy of the absorbed light to the A values, and two important observations emerge  

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The continuum treatment of dispersion forces due to Lifshitz [19,20] provides the appropriate analysis of retardation through quantum field theory. More recent analyses are more tractable and are described in some detail in several references [1,3,12,21,22],... 

Although the exact equations of state are known only in special cases, there are several usefid approximations collectively described as mean-field theories. The most widely known is van der Waals equation [2]... 

The parameters a and b are characteristic of the substance, and represent corrections to the ideal gas law dne to the attractive (dispersion) interactions between the atoms and the volnme they occupy dne to their repulsive cores. We will discnss van der Waals equation in some detail as a typical example of a mean-field theory. 

This is the well known equal areas mle derived by Maxwell [3], who enthusiastically publicized van der Waal s equation (see figure A2.3.3. The critical exponents for van der Waals equation are typical mean-field exponents a 0, p = 1/2, y = 1 and 8 = 3. This follows from the assumption, connnon to van der Waals equation and other mean-field theories, that the critical point is an analytic point about which the free energy and other themiodynamic properties can be expanded in a Taylor series. 

Table A2.3.4 simnnarizes the values of these critical exponents m two and tliree dimensions and the predictions of mean field theory.

As a prelude to discussing mean-field theory, we review the solution for non-interacting magnets by setting J = 0 in the Ising Flamiltonian. The PF... 

Fluctuations in the magnetization are ignored by mean-field theory and there is no correlation between neighbouring sites, so that... 

The neglect of fluctuations in mean-field theory implies that... 

Table A2.3.5 Critical temperatures predicted by mean-field theory (MFT) and the quasi-chemical (QC) approximation compared with the exact results.

An essential feature of mean-field theories is that the free energy is an analytical fiinction at the critical point. Landau [100] used this assumption, and the up-down symmetry of magnetic systems at zero field, to analyse their phase behaviour and detennine the mean-field critical exponents. It also suggests a way in which mean-field theory might be modified to confonn with experiment near the critical point, leading to a scaling law, first proposed by Widom [101], which has been experimentally verified. 

This implies that the critical exponent y = 1, whether the critical temperature is approached from above or below, but the amplitudes are different by a factor of 2, as seen in our earlier discussion of mean-field theory. The critical exponents are the classical values a = 0, p = 1/2, 5 = 3 and y = 1. 

The assumption that the free energy is analytic at the critical point leads to classical exponents. Deviations from this require tiiat this assumption be abandoned. In mean-field theory. 

Plotting r versus 1/n gives kTJqJ as the intercept and (kTJqJ)( -y) as the slope from which and y can be determined. Figure A2.3.29 illustrates the method for lattices in one, two and tliree dimensions and compares it with mean-field theory which is independent of the dimensionality. 

Weeks J D, Katsov K and Vollmayr K 1998 Roles of repulsive and attractive forces in determining the structure of non uniform liquids generalized mean field theory Phys. Rev. Lett. 81 4400... 

The integral under the heat capacity curve is an energy (or enthalpy as the case may be) and is more or less independent of the details of the model. The quasi-chemical treatment improved the heat capacity curve, making it sharper and narrower than the mean-field result, but it still remained finite at the critical point. Further improvements were made by Bethe with a second approximation, and by Kirkwood (1938). Figure A2.5.21 compares the various theoretical calculations [6]. These modifications lead to somewhat lower values of the critical temperature, which could be related to a flattening of the coexistence curve. Moreover, and perhaps more important, they show that a short-range order persists to higher temperatures, as it must because of the preference for unlike pairs the excess heat capacity shows a discontinuity, but it does not drop to zero as mean-field theories predict. Unfortunately these improvements are still analytic and in the vicinity of the critical point still yield a parabolic coexistence curve and a finite heat capacity just as the mean-field treatments do. 

Exponent values derived from experiments on fluids, binary alloys, and certain magnets differ substantially from all those derived from analytic (mean-field) theories. Flowever it is surprising that the experimental values appear to be the same from all these experiments, not only for different fluids and fluid mixtures, but indeed the same for the magnets and alloys as well (see section A2.5.5). 

Figure A2.5.26. Molar heat capacity C y of a van der Waals fluid as a fimction of temperature from mean-field theory (dotted line) from crossover theory (frill curve). Reproduced from [29] Kostrowicka Wyczalkowska A, Anisimov M A and Sengers J V 1999 Global crossover equation of state of a van der Waals fluid Fluid Phase Equilibria 158-160 532, figure 4, by pennission of Elsevier Science.

Levelt Sengers J M H 1999 Mean-field theories, their weaknesses and strength Fluid Phase Equilibria 158-160 3-17... 

Hamiltonian, but in practice one often begins with a phenomenological set of equations. The set of macrovariables are chosen to include the order parameter and all otlier slow variables to which it couples. Such slow variables are typically obtained from the consideration of the conservation laws and broken synnnetries of the system. The remaining degrees of freedom are assumed to vary on a much faster timescale and enter the phenomenological description as random themial noise. The resulting coupled nonlinear stochastic differential equations for such a chosen relevant set of macrovariables are collectively referred to as the Langevin field theory description. 

In a field theory description, the thennodynamic free energy /is generalized to a free energy fiinctional... 

The central quantity of interest in homogeneous nucleation is the nucleation rate J, which gives the number of droplets nucleated per unit volume per unit time for a given supersaturation. The free energy barrier is the dommant factor in detenuining J J depends on it exponentially. Thus, a small difference in the different model predictions for the barrier can lead to orders of magnitude differences in J. Similarly, experimental measurements of J are sensitive to the purity of the sample and to experimental conditions such as temperature. In modem field theories, J has a general fonu... 

Roothaan C C J 1960 Self-oonsistent field theory for open shells of eleotronio systems Rev. Mod. Phys. 32 179-85... 

The general fomi of the expansion is dictated by very general synnnetry considerations the specific coefficients for the example of a polymer blend can be derived from the self-consistent field theory. For a... 

Figure B3.6.5. Phase diagram of a ternary polymer blend consisting of two homopolymers, A and B, and a synnnetric AB diblock copolymer as calculated by self-consistent field theory. All species have the same chain length A and the figure displays a cut tlirough the phase prism at%N= 11 (which corresponds to weak segregation). The phase diagram contains two homopolymer-rich phases A and B, a synnnetric lamellar phase L and asynnnetric lamellar phases, which are rich in the A component or rich in the B component ig, respectively. From Janert and Schick [68].

Luckhurst G R, Zannoni C, Nordic P L and Segre U 1975 A molecular field theory for uniaxial nematic... 

Luckhurst G R 1985 Molecular field theories of nematics systems composed of uniaxial, biaxial or flexible molecules Nuclear Magnetic Resonance of Liquid Crystals ed J W Emsiey (Dordrecht Reidel)... 

The quantum phase factor is the exponential of an imaginary quantity (i times the phase), which multiplies into a wave function. Historically, a natural extension of this was proposed in the fonn of a gauge transformation, which both multiplies into and admixes different components of a multicomponent wave function [103]. The resulting gauge theories have become an essential tool of quantum field theories and provide (as already noted in the discussion of the YM field) the modem rationale of basic forces between elementary particles [67-70]. It has already been noted that gauge theories have also made notable impact on molecular properties, especially under conditions that the electronic... 

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