Theory self-consistent field

In the remainder of this section we describe the behavior of the coarse-grained model in the framework of the SCF theory. To allow for an analytical treatment, the SCF calculations use an even simpler model, which only reproduces the properties of the polymer model on long-length scales. 

To describe a compressible mixture of two molecules - solvent S and polymer P - we use two independent (segmental) number densities t/ s and cpp. The pressure depends on both densities and to capture the qualitative behavior we parameterize the equation of state by a virial expression that only retains terms up to third order [152]  

To investigate the properties of spatially inhomogeneous systems, we start from the grandcanonical partition function  

Equation (52) is compatible with the equation of state (50). Note that this interaction functional is strictly local, i.e., we implicitly assume the range of segmental interactions to be small compared to the width w of the liquid-vapor interface. 

Since the interactions are strictly local, it is not possible to represent the solvent as a structureless point particle. In the following we assume that the size of the solvent particle is comparable to the size of a polymer segment. To deal with solvent and polymer on the same footing, we describe the spatial distribution of the density of a solvent particle by that of a single polymer segment, i.e., we set Ns = 1 and bs = b. 

The following set of computer laboratories deals with self-consistent field theory as applied to simple atomic and molecular systems. 

This computer laboratory presents the simplest analytical (Roothaan-like) SCF calculation imaginable. The basis set is a linear combination of Slater-type orbitals truncated after the first term.i s The calculation was developed in response to a well-known series of related problems suggested by Slater. S jhe simplicity of the calculation ensures that the fundamentals of SCF theory will not be obscured by complicated mathematics and computer code. This calculation has been done using BASIC, Mathematica, and Mathcad. 

This laboratory is an extension of the previous exercise and is based on a paper by Snow and Bills.just as in the Hartree SCF calculation, the students are provided with detail theoretical background, a flowchart, and subroutines. They are asked to write the main program and execute a successful calculation. Students make the same types of comparisons between theory and experiment as were done in the Hartree numerical calculation. 

In this calculation, the trial wavefunction is a linear combination of the H Is and the Li 2s and 2p atomic orbitals. Theoretical background, a flowchart, and subroutines are again provided, and the students write the main program. After a successful run, the students compare the valence electron orbital energy with the first ionization energy of LiH. They also calculate the charge densities on Li and H and determine the dipole moment. This exercise is based on the more complete calculation of Karo and Olsen.i  

The marked increase in complexity which occurs when another electron is added to the hydrogen molecule ion to create the hydrogen molecule is seen in this computer lab. This exercise is based on a theoretical chemistry exercise 

The energy of an approximate wave function can be calculated as the expectation value of the ffamiltonian operator, divided by the norm of the wave function. 

SCFT today is one of the most commonly used tools in polymer science. SCFT is based on de Gennes-Edwards description of a polymer molecule as a flexible Gaussian chain combined with the Flory-Huggins local treatment of intermolecular interactions. Applications of SCFT include thermodynamics of block copolymers (Bates and Fredrickson, 1999 Matsen and Bates, 1996), adsorption of polymer chains on solid surfaces (Scheutjens and Fleer, 1979,1980), and calculation of interfacial tension in binary polymer blends compatibilized by block copolymers (Lyatskaya et al., 1996), among others. 

Over the past decade, SCFT was often applied to analyze the problem of particle dispersion in polymers (thermodynamics of nanocomposites). Vaia and Giannelis (1997a, 1997b) formulated a simple version of SCFT 

a is the lattice unit dimension, M is the number of lattice units per clay platelet (so that the product Ma2 = A is the total area of the platelet), a is the grafting density of surfactants, Xa/3 are the Flory-Huggins parameters between species a and (3, //, is the chemical potential of the zth component, and 0, is the excess amount of the zth component in the system. The density profiles of various species, j a(z), and conjugate fields, ua(z), are calculated as described below. Note that we introduce a separate species and component—voids (denoted as subscript v)—to account for density variation within the gallery. 

The bulk chemical potentials of all polymers and voids are described as 

In Equations (17a) and (17b), densities with superscript b refer to the equilibrium densities in the bulk (for all components and all species). The excess amount of each component, 0, is given by 

Having stated the limitations (non-relativistic Hamilton operator and the Bom-Oppenheimer approximation), we are ready to consider the electronic Schrodinger equation. It can only be solved exactly for the Hj molecule, and similar one-electron systems. In the general case we have to rely on approximate (numerical) methods. By neglecting relativistic effects, we also have to introduce electron spin as an ad hoc quantum effect. Each electron has a spin quantum number of 1 /2. In the presence of an 

To generate approximate solutions we will employ the variational principle, which states that any approximate wave function has an energy above or equal to the exact energy (see Appendix B for a proof). The equality holds only if the wave function is the exact function. By making a trial wave function containing a number of parameters, we cafT 

The energy of an approximate wave function can be calculated as the expectation 

For a normalized wave function the denominator is 1, and therefore Z e = ( J He ). The total electronic wave function must be antisynnnetric (change sign) with respect to the interchange of any two electron coordinates (since electrons are fermions. 

Let us in the following assume that we are interested in solving the electronic Schrddinger equation for a molecule. The one-electron functions are thus Molecular Orbitals (MO), which are given as a product of a spatial orbital times 

In the mid-1960s EdwcU-ds pointed out a remarkable analogy between the problem of interacting polymers and the classical problem of interacting 

We know that an ideal polymer chain is a random walk. Consider a distribution function q r, r, t), which tells us the probability that a chain that is t steps long has started at a position r and finished at a position r. We know that a random walker in free space has a distribution function that obeys a diffusion equation, so for an ideal, isolated chain (with no excluded volume interaction) we can write 

The diffusion coefficient is simply related to the square of the statistical step 

Now suppose the random walk is not in free space, but is affected by a spatially varying potential U(r). This now makes the statistical weights of various possible steps in the random walk unequal via a Boltzmann factor and the effect of this is to modify the diffusion equation thus  

This is the basic equation of the self-consistent field theory the reader may notice that it has the same form as Schrodinger s equation. This is the basis of the analogy between polymer chain conformations in interacting systems and the properties of interacting electrons. 

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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].

Many problems in force field investigations arise from the calculation of Coulomb interactions with fixed charges, thereby neglecting possible mutual polarization. With that obvious drawback in mind, Ulrich Sternberg developed the COSMOS (Computer Simulation of Molecular Structures) force field [30], which extends a classical molecular mechanics force field by serai-empirical charge calculation based on bond polarization theory [31, 32]. This approach has the advantage that the atomic charges depend on the three-dimensional structure of the molecule. Parts of the functional form of COSMOS were taken from the PIMM force field of Lindner et al., which combines self-consistent field theory for r-orbitals ( nr-SCF) with molecular mechanics [33, 34]. 

Constanciel R and R Contreras 1984. Self-Consistent Field Theory of Solvent Effects Representation by Continuum Models - Introduction of Desolvation Contribution. Theoretica Chimica Acta 65 1-11. 

The simplest model of polymers comprises random and self-avoiding walks on lattices [11,45,46]. These models are used in analytical studies [2,4], in particular in the numerical implementation of the self-consistent field theory [4] and in studies of adsorption of polymers [35,47-50] and melts confined between walls [24,51,52]. 

We close these introductory remarks with a few comments on the methods which are actually used to study these models. They will for the most part be mentioned only very briefly. In the rest of this chapter, we shall focus mainly on computer simulations. Even those will not be explained in detail, for the simple reason that the models are too different and the simulation methods too many. Rather, we refer the reader to the available textbooks on simulation methods, e.g.. Ref. 32-35, and discuss only a few technical aspects here. In the case of atomistically realistic models, simulations are indeed the only possible way to approach these systems. Idealized microscopic models have usually been explored extensively by mean field methods. Even those can become quite involved for complex models, especially for chain models. One particularly popular and successful method to deal with chain molecules has been the self-consistent field theory. In a nutshell, it treats chains as random walks in a position-dependent chemical potential, which depends in turn on the conformational distributions of the chains in... 

McWeeny, R., Proc. Roy. Soc. [London) A235, 496, (i) The density matrix in self-consistent field theory. I. Iterative construction of the density matrix." Beryllium atom is studied. Steepest descent method is described. 

G. Berthier, M. Defranceschi, J. Delhalle in Self-Consistent Field Theory and Applications. R. Carbo, M. Klobukowski Eds, Elsevier, Amsterdam 1990, "Studies in Physical and Theoretical Chemistry ", n 70, p. 387... 

MUller, M. and Schmid, F. Incorporating Fluctuations and Dynamics in Self-Consistent Field Theories for Polymer Blends. Vol. 185, pp. 1-58. 

This chapter is concerned with the application of liquid state methods to the behavior of polymers at surfaces. The focus is on computer simulation and liquid state theories for the structure of continuous-space or off-lattice models of polymers near surfaces. The first computer simulations of off-lattice models of polymers at surfaces appeared in the late 1980s, and the first theory was reported in 1991. Since then there have been many theoretical and simulation studies on a number of polymer models using a variety of techniques. This chapter does not address or discuss the considerable body of literature on the adsorption of a single chain to a surface, the scaling behavior of polymers confined to narrow spaces, or self-consistent field theories and simulations of lattice models of polymers. The interested reader is instead guided to review articles [9-11] and books [12-15] that cover these topics. 

The density functional theory has the stmcture of a self-consistent field theory where the density profile is obtained from a simulation of a single chain in the... 

The obvious disadvantage of this simple LG model is the necessity to cut off the infinite expansion (26) at some order, while no rigorous justification of doing that can be found. In addition, evaluation of the vertex function for all possible zero combinations of the reciprocal wave vectors becomes very awkward for low symmetries. Instead of evaluating the partition function in the saddle point, the minimization of the free energy can be done within the self-consistent field theory (SCFT) [38 -1]. Using the integral representation of the delta functionals, the total partition function, Z [Eq. (22)], can be written as... 

R. Constanciel and R. Contreras, Self-consistent field theory of solvent effects representation by continuum models-introduction of desolvation contribution, Theor. Chim. Acta 65 1 (1984). 

Figure 11. A schematic representation of the mean-field approximation, a central issue in the self-consistent-field theory. The arrows symbolically represent the lipid molecules. The head of the arrow is the hydrophilic part and the line is the hydrophobic tail. On the left a two-dimensional representation of a disordered bilayer is given. One of the lipids has been selected, as shown by the box around it. The same molecule is depicted on the right. The bilayer is depicted schematically by two horizontal lines. The centre of the bilayer is at z = 0. These lines are to guide the eye the membrane thickness is not preassumed, but is the result of the calculations. Both the potential energy felt by the head groups and that of the tail segments are indicated. We note that in the detailed models the self-consistent potential profiles are of course much more detailed than shown in this graph...

Most of these developments may be applied most directly within the framework of the isolated molecule method, in which the reactivity indices are the charges and self-polarizabilities of the unperturbed ground state of a given molecule calculations based on the localization model (e.g. Nesbet, 1962) have made less progress, and will not be considered. It is therefore natural to enquire whether indices similar to and tt,, in Hiickel theory can still be defined, and calculated more precisely, in self-consistent field theory. The obvious questions are... 

Scaling theory Self-consistent field theory ... 

Theoretical calculations of minimum-energy structures and thermodynamic terms using self-consistent field theory with thermodynamic and solvation corrections concluded that the cyclization of l-hydroxy-8-(acetylamino)naphtha-lene 1 to give 2-methylnaphth[l,8-r7,< ][l,3]oxazine 2 with the liberation of water was much less favorable (AG = —2.0kj moP, A/7 = 4-31.0kj mol , and TAS = 4-33.1 kjmoP at 298.2 K) in the gas phase than the corresponding ring closure of l-amino-8-(acetylamino)naphthalene, which was in qualitative agreement with experimental observations for the reactions in solution <1998J(P2)635>. 

J. Paldus, Hartree-Fock Stability and Symmetry Breaking. In R. Garbo and M. Klobukowski (Eds.) Self-Consistent Field Theory and Applications (Elsevier, Amsterdam, 1990), pp. 1-45. 

EXCITED STATE SELF-CONSISTENT FIELD THEORY USING EVEN-TEMPERED PRIMITIVE GAUSSIAN BASIS SETS... 

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