Polymer 
Consequently, it is generally not useful to focus on the lowest energy structure alone if one is interested in the properties of a polymer in the melt or in solution. (It might be appropriate for the crystalline state, however.) Rather, one must deal with the statistical behavior of a chain, taking into account all the accessible possibilities for the chain conformation. In this discussion, the focus is on the structural properties of chainstheir size and stiffness. These are the simplest properties both to visualize and to calculate, but one can also calculate dipole moments, optical properties, and other quantities by means of Rotational Isomeric State (RIS) theory.
A measure of polymer stiffness, indirectly related to polymer shape, is the persistence length. This can be defined in various ways. The persistence length may be defined as the average sum of the projections of all bond vectors onto the first bond of a chain. Alternatively, it may be defined as the projection of all succeeding bonds (including the bond itself) onto an internal bond of the chain. The persistence length is a measure of the distance over which a chain retains "memory" of its initial direction. A flexible chain tends to lose all memory of its direction after a short distance and thus tends to have a small value of the persistence length. By contrast, a stiff chain has a comparatively large value. Note that the "stiffness" defined here is a static, equilibrium stiffness, as opposed to a dynamic stiffness. (The latter depends on the barriers to rotations of the chain backbone, not just the relative energies of the various rotational states of the chain.)
Random walk models are not limited to those on lattices. An example of an offlattice random walk model is the freely jointed chain. Such a chain consists of n segments, joined at vertices with unrestricted bond angles. (It is as if a pivot is at each vertex, allowing the connected segments to swing about freely; hence the term freely jointed.) The chain statistics for a freely jointed chain are the same as for a random walk on a lattice; i.e., r^{2} = nl^{2} and s^{2} = r^{2}/6. Note that because all succeeding bonds are uncorrelated with the first, the persistence length is equal to the bond length for both the random walk on a lattice and the freely jointed chain. That is, these chains have no stiffness beyond the trivial stiffness of a single bond.
It is obvious that the above models omit all chemical details of chains. In addition, they make another major approximation. They neglect excluded volume interactionsthat is, they neglect the fact that a real polymer chain cannot intersect with itself. For a chain in dilute solution under theta conditions, this approximation is a good one. It is also valid for a chain in the melt and for relatively short or stiff chains. (Of course, it is not literally true that the chain can intersect itself under these conditions. But for the purpose of computing conformational averages, the chain can be treated as if this were possible.) In other cases, such as in a good solvent, excluded volume can have a dramatic effect on the chain statistics. In particular, r^{2} scales roughly as n^{1.2}, rather than n^{1}, for a chain whose excluded volume interactions are not negligible. Thus, an excluded volume chain is significantly larger than a theta chain. Excluded volume interactions make theoretical treatment of a chain difficult, and in order to predict properties of realistic chains with excluded volume, explicit simulation (Monte Carlo or molecular dynamics) must be performed. RIS theory treats theta chains only. Although doing so is computationally expensive, the RIS Monte Carlo method can be modified to treat excluded volume in an approximate fashion. (This has been done in the RIS module.) When excluded volume interactions are negligible, a chain's dimensions are referred to as unperturbed dimensions and are usually designated by the subscript 0 (e.g., r^{2}_{0}).
RIS theory forms the basis for Flory's book, Statistical Mechanics of Chain Molecules (originally published in 1969, and republished in 1989), and for a later review article (Flory 1974) that reformulated the theory in a simpler and more general form. Refer to these sources for the details of the method, which is only outlined here.
Like most computationally tractable theories of polymer chains, RIS theory neglects long range excluded volume interactions. (Here, long range refers to the separation along the contour of the chain, not to the separation in 3D space.) The theory does account for short range excluded volume effects and for chain stiffness, however. Most importantly, unlike simplified random walk models, it accounts for the detailed chemical structure of a polymer chain.
The RIS approach takes advantage of the fact that in determining the overall conformation of a linear (i.e., unbranched) chain, the most important degrees of freedom are torsional rotations about the bonds of the backbone. The force constants for other degrees of freedom, such as bond bending and stretching, are so large that bond angles and lengths do not vary greatly from their equilibrium values. Consequently, such motions generally make little contribution to the gross conformation of a chain. In the RIS scheme, all degrees of freedom except torsions in the backbone are explicitly neglected. (The effects of these other degrees of freedom may be implicitly taken into account in determining the statistical weights, however.)
A torsion angle may in principle adopt a continuum of values from 180º to 180º. For the purpose of determining conformational statistics, the rotational isomeric state approximation treats a torsion angle in a chain's backbone as if it can adopt only a discrete subset of these values. The justification for this is made clear by inspecting a plot of the potential energy as a function of the dihedral angle. Such a plot typically contains a small number of distinct minima, separated by relatively high energy barriers. In polyethylene, for example, a typical RIS model considers only three values for any given torsion angle, corresponding to one trans and two gauche minima in the energy profile. Although only a single value of the dihedral angle () for each state enters the calculation, deviations from this value are not disregarded but are implicitly included in the weights.
In the RIS scheme, any conformational property, A, is assumed to be a function only of the dihedral angles in the chain backbone:
For certain properties A, the value of the property for a given conformation may be expressed:
Writing A as a product of generator matrices gives its value for a single conformation, but to obtain conformational statistics it is necessary to average over all possible conformations. That is, one must perform the sums over all the _{i} with each term weighted by the exponential Boltzmann factor. To see how this is done, it is convenient to first address the calculation of the configurational partition function. This will introduce the statistical weights matrices, which will be used in performing the sums over the _{i}.
In the RIS scheme, the configurational partition function may be written:
For independent bond rotations, the weight associated with a given conformation is:
For an internal bond pair in polyethylene at 140ºC, the statistical weights matrix is:
The energies that are used to determine the statistical weights may be obtained from contour maps of the energy as a function of two adjacent torsion angles in a small fragment of a chain. In principle, the "energy" in such maps should be a potential of mean force, obtained by averaging over all other degrees of freedom in the problem. In addition, some or all of the statistical weights may be optimized by constraining certain RIS results (e.g., characteristic ratios) to agree with experiment. These same statistical weights may then be used to calculate other properties for which experimental data are not available.
Given the above formulation, the average property A may be expressed:
is the diagonal array of generator matrices F_{i} for the rotational states 1 ... _{i}, and E_{s} is the identity matrix of the same order as F_{i}. The subscript 0 has been appended to A because neglect of long range forces is implied by the assumption that bond energies depend on nearest neighbor interactions alone.
Although a fair amount of algebra (most of which has not been shown in this introductory presentation) is required to obtain the above result, the simplicity of this result should be noted. In the RIS model, the extremely complicated problem of averaging a property over all possible chain conformations is reduced to serial multiplication of matrices. To summarize, the RIS model gives the conformational statistics of unperturbed, realistic polymer chains by reducing complicated integrals to simple matrix multiplication.
Some of the more common properties of unperturbed chains that may be calculated using RIS theory are the following: mean square endtoend distance (and consequently, the characteristic ratio), radius of gyration, persistence length, mean square dipole moment, polarization anisotropy, stressoptical coefficient, molar Kerr constant (related to electric birefringence), higher moments of r and s (e.g., r^{4}), and macrocyclization and epimerization equilibria.
Although a RIS Monte Carlo (RIS MC) calculation is computationally more intensive than a standard RIS calculation, the theory behind it is simpler in some ways. RIS MC makes use of the statistical weights described previously, but avoids the need to compute generator matrices for each property of interest.
In performing a RIS MC calculation, chain configurations (or conformationsthese terms are used interchangeably here) are generated with the same probability distribution as that implicit in the statistical weights. A given property or set of properties is calculated for each conformation, and the results are averaged over a large number of these.
The normalized probability of occurrence for a given chain conformation in the RIS scheme (see Eq. 492 and Eq. 495) is:
The RIS MC procedure generates a chain conformation by beginning at the tail of the chain and setting backbone rotational states onebyone until the head of the chain is reached. To be able to select such states requires that singlebond a priori probabilities and bond pair conditional probabilities be calculated. These are defined as follows. The probability of bond i being in state , irrespective of the states of all other bonds, is referred to as the singlebond a priori probability. The probability of bond i being in state , given that bond (i  1) is in state ', is the bond pair conditional probability. These probabilities are determined by the statistical weights in such a way that the probability of generating a conformation using RIS MC is the same as that given in Eq. 500.
Specifically, the single bond a priori probability is given by:
To obtain conditional bondpair probabilities, we first need a priori pair probabilities, which are given by expressions of the type:
For every conformation generated by the Monte Carlo process, appropriate data (e.g., endtoend distance, radius of gyration) are calculated and running averages or distribution functions are maintained. The process of generating a conformation is repeated until a preselected total number has been reached.
Among the quantities calculable from RIS Monte Carlo are the following:
The eigenvalues of this tensor give the principal components of s^{2} in three mutually perpendicular directions. The probability distribution of the square roots of these eigenvalues (denoted p(S_{1}, S_{2}, S_{3})) indicates the average shape of the chain; for example, the extent to which it tends to be prolate as opposed to oblate. When represented as a 3D contour plot, p(S_{1}, S_{2}, S_{3}) gives a vivid illustration of the chain's overall shape.

The total scattering function for a molecule is a sum of contributions from all different types of scattering centers. In a RIS Monte Carlo calculation, each of these separate contributions may be computed as well. Suppose A and B are two types of scattering centers in a molecule. Then the total S(k) is represented in terms of partial S(k)'s as follows (assuming the scattering amplitude is the same for both centers):
Both the conventional RIS and RIS MC methods require that statistical weights exist for the polymer of interest. There is a third method"RIS" Metropolis Monte Carlo (RMMC) simulationthat does not require statistical weights. Instead, RMMC uses a forcefield (such as CVFF or PCFF) to determine the conformational properties of a chain. Because it does not use statistical weights (or even discrete torsion angles), RMMC is not, strictly speaking, a rotational isomeric state method. However, it shares several assumptions with RIS theory, and is used to compute the same types of properties as RIS. In this sense, it is closely related to RIS theory.
A stepbystep presentation of the RMMC procedure appears later in the The RMMC algorithm section. First, if you wish to decide which method is best for your polymer, consider these major differences between the RIS MC and RMMC methods:
As in traditional RIS methods, only torsional degrees of freedom are considered in determining a chain's conformation; bond lengths and angles are fixed. Unlike these methods, RMMC allows torsion angles to vary continuously; it does not impose the assumption of discrete rotational states.
As its name indicates, RMMC is a Metropolis (1953) Monte Carlo method. (This can be contrasted to the Markovian approach used in a RIS MC calculation to build independent chains.) In a Metropolis simulation of a polymer, one begins with a chain in an arbitrary conformation. A Monte Carlo step consists of making a small change to that conformatione.g., by rotating a bondand then deciding whether or not to retain that change, based on the temperature and the energy of the new conformation relative to the old one. This process is repeated many times in order to yield a set of conformations characteristic of that chain at the specified temperature.
In outline, a RMMC simulation proceeds as follows:
1. Perform an energy minimization on the molecule (so that bond lengths and angles adopt reasonable values).
2. Randomly select a rotatable backbone bond.
3. Select a random torsion value for this bond between 180 and +180 degrees.
4. Rotate the bond to its new torsion value and compute the new energy of the chain.
5. Generate a random number, R, between 0 and 1. If exp[(E_{new} E_{old})/kT] > R, keep the new torsion value. Otherwise, restore the old value.
6. After enough steps have been done that the chain is equilibrated: compute the properties of the chain conformation (e.g., r^{2}, s^{2}), and update the running averages of these properties.
7. Repeat from step (2) until the desired number of iterations has been performed.
In a Monte Carlo simulation that incorporates constraints (such as fixed bond lengths and angles) and uses internal coordinates (rather than Cartesian coordinates) in making its moves, care must be taken that the conformations are sampled correctly. Simulations may be performed with constraints that are regarded as "rigid" or as "flexible." In the rigid case, the momenta conjugate to the constrained coordinates are neglected. In the flexible case, it is imagined that the force constants for the constrained coordinates are so large that the coordinates do not move significantly from their equilibrium values, but that the conjugate momenta are activated. It is hard to say a priori which is a more reasonable approximation for a polymer. In a real polymer, bond lengths and angles are, of course, not rigidly constrained. On the other hand, according to quantum mechanics, very stiff degrees of freedom are not activated at ordinary temperatures. Go and Scheraga (1976) argue that, in practice, flexible constraints should be more quantitatively accurate.
It should be emphasized that the distinction between "rigid" and "flexible" constraints is in the assumptions behind the physical model. Thus, even in the "flexible" case, the constrained coordinates do not move during the simulation.
A flexible constraint requires no special treatment in a Monte Carlo simulation. This is because the configuration integral, as expressed in terms of torsional degrees of freedom, contains a Jacobianlike term that is independent of these degrees of freedom. Thus, this term can be moved outside the integral, and is divided out of any conformational averages. In the rigid case, this is not true, and in principle the Boltzmann factor used in the Metropolis acceptance criterion needs to be multiplied by an extra term (Fixman 1974). In practice, this extra term has been found to change the conformational statistics only slightly (Almarza et al. 1990). For simplicity, the flexible constraint assumption, with no extra terms in the Boltzmann factor, is made for the RMMC algorithm. Any errors introduced by this approach are likely to be much smaller than those due to other approximations in the RMMC algorithmsuch as approximating the potential of mean force by a forcefield with a truncated interaction range. For more on the issue of constraints in simulations, see Fixman (1974), Allen and Tildesley (1987), and Almarza et al. (1990).
In the database you receive with the polymer software, all data have been obtained from the literature. (The references are given in the statistical weights data file; see also Appendix C, File Formats.) Parameters for the following polymers and only these polymers are included in the database:
The following are special cases of groups: A ring or fused ring in the backbone is defined to be a single group, irrespective of the number of atoms it contains. A cis double bond is regarded as a single group, connected by virtual bonds to neighboring groups. On the other hand, a trans double bond is regarded as a sequence of two groups, which in a RIS calculation, are treated as comprising a bond with only one rotational state. A triple bond or any sequence of more than one multiple bond is defined to be a single group.
The RIS module maintains a database of known groups. For the module to identify a chain properly, the groups making up the chain must be in this database. When the chain sequencer is presented with a polymer, it attempts to find a match between the groups in the chain's backbone and known groups. In attempting to find a match, it accounts only for the connectivity between atoms, not for their positions in 3D space, with one exception. The exception is consideration for the stereochemical orientation of pseudoasymmetric centers in the backbone of a tactic chain. In this way, account is made for tacticity. Account is not made for distinctions based on stereochemistry in side groups.
It is quite easy to add new groups to the database of known groups, but the bond properties and statistical weights databases must be updated as well for RIS calculations to be performed on a new polymer (see the Adding a new group and Adding new properties sections).
For the purpose of specifying bond properties, a bond is defined by two connected groups. For example, an interior bond of polyethylene is defined by the group sequence methylene, methylene.
In performing RIS calculations, one end of a chain is chosen to be the tail and the other to be the head. However, this designation is completely arbitrary, and RIS theory yields the same conformational properties irrespective of which chain end is chosen as the tail. Because of the head/tail arbitrariness in a RIS calculation, if A and B are two different groups, and you add properties for an AB bond to the database, you should also enter properties for a BA bond.
Because of the pentane effect, a bond pair is defined by a sequence of five groups, for the purpose of assigning statistical weights. For example, an interior bond pair in polyethylene is defined by the sequence methylene, methylene, methylene, methylene, methylene. For syndiotactic poly(vinyl chloride), one type of internal bond pair has the sequence methylene, dchloromethylene, methylene, lchloromethylene, methylene. For a RIS calculation to be performed on a particular polymer, all distinct sequences of five groups in the chain's backbone must have weights stored in the database. (However, see the information in Appendix C, File Formats, on using wildcards in specifying bond sequences. Also, see below for dealing with terminal and nearterminal bonds.)
A statistical weight for a bond pair is stored in the database as a prefactor and an energy, such that the value of the weight at a given temperature T is given by:
The calculation of the statistical weights matrix for a given bond pair in a polymer chain entails the following steps. First, a short chain segment (which commonly has a length of four skeletal bonds including the bond pair in question) is identified and extracted from the polymer chain. Consider, for example, the following chain segment which has been extracted from polytetrafluoroethylene (PTFE):
The second step is the determination of the total energy of this segment as a function of the two torsion angles _{1} and _{2}. This is accomplished by running Discover to find the minimum energy corresponding to each fixed pair of values of the rotational angles.
Having obtained energy E^{t}(_{1},_{2})^{2} as a function of the two torsion angles, the third step is the construction of a contour map showing the lines of constant energy on a two dimensional graph. The energy map for the PTFE segment shown above is given in Figure 56.

Next, the map is analyzed to locate the energy minima. The purpose of this analysis is to determine the discrete states (i.e., angle values) to be used in the RIS approximation. To this end, the map is divided into a set of regions. In most common situations, each local minimum would be surrounded by a distinct region. This is exemplified in Figure 57 for the PTFE segment.

When two or more minima are within close proximity to each other, however, it makes sense to group them together within a single region which then allows the representation of the minima in question together as a single state on the map.
The application of these ideas to another map (which is included below as an example) is illustrated in Figure 58. Note that there are 11 minima and 9 regions on this map. Two pairs of minima have been formed by lumping individual minima which are close to each other into pairs, each pair being surrounded by a single region.

Before discussing how these regions are employed in the calculation of statistical weights, another crucial step of the calculations is first explained here.
The energy values obtained for the PTFE segment above include all firstorder and secondorder interactions in the segment. Those interactions that depend on the value of exactly one torsion angle are termed firstorder interactions. The interactions that depend on the values of exactly two (usually adjacent in the backbone) torsion angles are termed secondorder interactions (Flory 1989).
For the PTFE segment shown earlier, the interaction between C1 and C4, for example, is a firstorder interaction. The energy changes due to the attractive and repulsive forces between C1 and C4 depend only on the value of the first torsion angle, _{1}, and not on the second angle, _{2}. On the other hand, the interaction between C1 and C5 depends on both of the torsion angles, since the distance between C1 and C5 depends on both _{1} and _{2}. Thus, the C1C5 interaction is a secondorder interaction.
By convention, the statistical weight matrix for a given bond pair should be based on all the secondorder interactions in the segment and only those firstorder interactions that are dependent on the second torsion angle _{2}. In the RIS approximation, the total conformational energy of a polymer chain is expressed as the sum of the energies of its segments. Since the energy changes due to the rotation of the first bond (i.e., the firstorder interactions dependent on _{1}) are accounted for in the calculations for the neighboring segment (which includes the first bond of the present segment as its second bond) these changes should be excluded from the calculations for the present segment to avoid double counting them.
As was pointed out earlier, the energy map constructed includes all first and secondorder interactions within the segment in question. To calculate the statistical weights correctly from this map, the energy due to the firstorder interactions dependent on _{1} must be determined and subtracted. To this end, a second set of Discover calculations is carried out on the following chopped segment extracted from the original polymer segment.
Note that _{2} is not defined on this molecule since there are no side groups attached to C4. Therefore, all first and secondorder interactions involving _{2} are absent from this smaller segment. In this model, only the firstorder interactions (such as the C1C4 interaction) that depend on _{1} are included. Discover is used to calculate the energy of the chopped segment as a function of the first torsion angle, i.e., E^{seg}(_{1}). The net energy corresponding to a value of _{1} is obtained by subtracting the chopped segment energy from the total energy at the same value of _{1}:
The next step is the calculation of average angles _{1} and _{2}, the average energy E, and the partition function z for each region. The following integrals are evaluated to obtain these quantities:
The next step is the determination of the number and locations of the states, i.e., _{1} and _{2} values that will be employed as the discrete states in the RIS approximation. If all _{1} and _{2} values were distinct (that is, if i j then _{1}i _{1}j, and the same condition holds for _{2} ), then there would be as many discrete _{1}and _{2} values as there are regions. If the number of regions is Nr, then the dimension of the resulting weight matrix would be N_{r} XN_{r}. Frequently, however, two or more regions may have the same, or similar, _{1} or _{2} values. In such cases, it is desirable to represent both regions using average _{1}, _{2} values. If the _{1} values of certain regions are to be combined so that they are all represented by a single _{1}_{combined} value, this combined average angle is determined as follows:
Let the number of _{1} values (obtained in the manner described above) be N_{1}, and the number of _{2} values be N_{2} (note that N_{1} N_{r} and N_{2 }N_{r}). Further, let these angles (some of which may have been obtained by combining the average angles of two or more regions) be denoted by _{1}^{c}_{i} and _{2}^{c}_{j}, 1 i N_{1}; 1 j N_{2}. These angles are indexed such that they are in ascending order, for example:
If there is no region with the average angles _{1}^{c}_{i} and _{2}^{c}_{j,} then:
In performing RIS calculations on vinyl and other tactic chains, many authors have taken advantage of symmetries inherent in such chains to simplify their calculations. For example, in some treatments of vinyl chains, the sign convention for dihedral angles differs for different types of bonds in the chains. (See Chapter VI of Flory 1989, and Flory et al. 1974.) Because the core of the RIS module is general, it does not take advantage of such simplifications and uses the same sign convention and same form of transformation matrix for all bonds of a chain. The sign convention is that used in Flory's book, Chapter I (1989). (Positive rotations are measured in a righthanded sense.) If you obtain weights for vinyl chains from published results, you must be careful to confirm that angles are of the proper sign before you enter them.
Even if you consider only isotactic chains, you must enter parameters for both dcenters and lcenters in the chain. This is because the selection of one end of a chain as tail and the other as head is arbitrary, and therefore unpredictable in advance. This means, for example, that an isotactic chain may be regarded equally well as a sequence of dcenters or lcenters by the RIS module.
There are two parametersMin Bonds and Max Bondsthat determine the interacting pairs of atoms for the purpose of calculating nonbond (van der Waals and Coulomb) energies. Nonbond energies are not computed for atoms closer than Min Bonds bonds away from each other. (See Figure 59.) The usual value for Min Bonds is 3. Nonbond energies are also neglected for atoms further than Max Bonds bonds away from each other. Reasonable values for Max Bonds range from 4 to about 6 for polymer chains in theta conditions. (Larger values of Max Bonds may greatly increase CPU and memory requirements, although some excluded volume effects might, in principle, be treated in this way.)

Whether articulated side groups are treated as flexible in a RMMC simulation is determined by whether the side group atoms have their backbone flags set. (The Sequencer/Set_Backbone command can be used to set these flags. It is best to do so on the repeat units before building the polymer.) In principle, it is more realistic to treat side groups as flexible rather than rigid. However, doing so increases the computation time and might not be necessary if the groups are small (as, for example, in polypropylene).
The "energy scaling factor" is a number that multiplies all computed energies. If experimental data are available, this factor can be used to refine the properties calculated by RMMC so that they match experimental properties as closely as possible. The same factor can then be used for calculations on related chains for which no experimental data exist.
The reason for the relatively large number of parameters controlling the energy calculation is as follows. A RMMC simulation is performed on an isolated polymer chain. Yet, the properties desired are those for a chain in solution or in the melt. If solvent molecules or other chain molecules were present in the simulation, then a high quality forcefield such as PCFF should be capable of predicting the correct properties. Because these other molecules are not explicitly present, their effects must be mimicked by altering the way the energy is computed. (In the language of statistical mechanics, it is not "bare" interaction energies but potentials of mean force (PMF) that determine the polymer conformations in a solvent or in the presence of other chains. See McQuarrie (1976) and Mattice and Suter (1994) for more on the PMF concept. The goal in a RMMC simulation is to have the computed energy come as close to the potential of mean force as possible.)
Using Max Bonds to limit the range of nonbond interactions is a way of mimicking theta conditions. However, there is no known a priori way to determine the ideal value of Max Bonds for a particular polymer. Thus, calculations with differing values of this parameter should be performed and the results evaluated according to their reasonableness or agreement with established data.
The dielectric constant provides another way of mimicking the presence of solvent. But it must be kept in mind that, at a molecular level, a solvent is not a dielectric continuum, so that this too is an approximation. Consequently, the best value for the dielectric constant in a RMMC calculation might not be the same as the system's bulk dielectric constant. In the event that variation of Max Bonds and the dielectric constant within reasonable limits does not give adequate results, the energy scaling parameter is available as a last resort.
In addition to the core pulldowns in the top menu bar, the RIS module adds the Sequencer, RIS_Compute, RIS_Monte_Carlo, NMR_System, Database, Stat_Weights, Graph, and Background_Job pulldowns to the lower menu bar.
Please see Chapter 21, Sequencer Pulldown, for more information on the commands within the Sequencer pulldown.
Calculations may be performed on a single chain (of fixed or variable length), or on each chain in an assembly of chains, with averaged results compiled automatically.
Calculations may be performed on a single chain of fixed length, or on each chain in an assembly of chains, with averaged results compiled automatically.
Please see Chapter 19, Graph Pulldown for more information.
Please see Chapter 15, Background_Job Pulldown for more information.
The Substitute command is used to perform a group substitution. This is accomplished by identifying the atom to change and the name of the group to be associated with that atom. As groups are substituted, the labels marking the unknown atoms are removed. When all unknown groups are substituted, no labels remain. You may then carry out the RIS calculation by performing step 4 above.
Even if you use the Substitute command successfully, you may find that statistical weights for your chain are unknown. This means that even though all the groups in your chain are known, there are sequences of groups for which weights are not known. If this happens, you must update the properties and weights databases to perform calculations on your molecule. Before doing so, you must first add any new groups to the database of known groups. Statistical weights may be obtained from the literature or estimated from energy contour plots, using the commands in the Stat_Weights pulldown.
The bond pair and the corresponding rotational angles in the cis arrangement can be defined as follows:
The calculations carried out on this segment will give the desired statistical weights.
As pointed out above, the RIS method treats a trans double bond as a onestate bond. The statistical weight matrix for the angle pair _{i}_{1} and _{i} requires no calculation as it is a column matrix consisting of 1's. Its dimension is determined by the number of columns of the matrix for the previous bond pair _{i}_{2} and _{i}_{1}. The matrix corresponding to the bond pair _{i} and _{i}_{+1} is a row matrix. The calculation of this matrix requires conformational energy calculations based on the rotation of a single angle (i.e., _{i}_{+1}) only.
Consider the following structure within the backbone of a chain:
The two backbone bonds connecting the ring to the rest of the chain are "replaced" by two virtual bonds as shown in the illustration above. That is, this ring is treated in exactly the same manner as a cis double bond in the backbone. The statistical weight matrix for the virtual bond pair in question can be obtained by defining the rotational angles as follows:
The following ring structure is analogous to a trans double bond in the backbone:
Here, a onestate virtual bond (i.e., nonrotatable) is placed across the ring cluster as shown in the illustration. The matrix corresponding to the angle pair _{i} and _{i}_{+1} is a row matrix which requires singleangle conformational calculations. Although the Stat_Weights tools can be used to calculate this matrix automatically, note that the Sequencer currently does not treat such structures properly. You may, however, treat fused rings of this type by editing the Sequencer output file.
Note: You must have write privileges for the system groups database to be able to add new groups. If not, an error message is returned when you try to add a new group. If this happens, see your system manager regarding write privileges for the database, or make your own personal copy of the system database as described in the Sequencer Tutorial.
> cp $BIOSYM/data/polymer/ris/system.prop user.propBecause you need to add new weights as well, type:
> cp $BIOSYM/data/polymer/ris/system.statwt user.statwtto get a local copy of the system weights file. (Later, when performing calculations using data in these files, you will use the RIS_Compute/Files command to select these files.)
Once a local properties file is in place, use the Database/Properties command to add new entries to this file. Specify the particular file you want to modify (in case there is more than one) and the following properties: bond length (in Å^{3}), bond dipole moment (in Debye), elements of the anisotropic optical polarizability tensor (in Å), and elements of the anisotropic static polarizability tensor (in Å^{3}). Of these properties, only the bond length is essential. If you lack data on the other properties, enter 0 for each of them, but keep in mind that quantities dependent on them (mean square dipole moment and optical properties) will be calculated incorrectly.
1  2  3  4  
1  PHIk  
2  A1k, EPS1k  
3  A2k, EPS2k  
4  A3k, EPS3k 
1  2  3  4  
1  PHIk  0.0  120.0  120.0 
2  A1k, EPS1k  
3  A2k, EPS2k  
4  A3k, EPS3k 
1  2  3  4  
1  PHIk  0.0  120.0  120.0 
2  A1k, EPS1k  1.0, 1.0  1.0, 0.5  1.0, 0.5 
3  A2k, EPS2k  1.0, 0.0  1.0, 2.5  1.0, 0.5 
4  A3k, EPS3k  1.0, 0.0  1.0, 0.5  1.0, 2.5 
Conceptually, a statistical weights matrix is associated with the second bond of a bond pair; that is, the bond between Group 3 and Group 4 when you define the bond pair by a group sequence. The first bond of a chain has an imaginary zeroth bond that precedes it. The first bond is specified as follows (e.g., for polyethylene):
Group 1 = NONE
Group 2 = NONE
Group 3 = methyl
Group 4 = methylene
Group 5 = methylene
Note, first, that you use NONE to specify nonexistent groups off the end of the chain, and second, that it is a methyl (CH_{3}) rather than a methylene (CH_{2}) group that terminates the chain. In Flory's convention, the statistical weights matrix for the first bond of a chain is a 1 X _{1} vector of the form:
The weights matrices of the second and third bonds of a chain do not take on any special form. However, for the RIS module to be able to break down a chain into contiguous sequences of five groups and thereby assign weights for the complete chain, the sequences defining these bonds must be specified. For example:
Group 1 = NONE
Group 2 = methyl
Group 3 = methylene
Group 4 = methylene
Group 5 = methylene
Group 1 = methyl
Group 2 = methylene
Group 3 = methylene
Group 4 = methylene
Group 5 = methylene
If you want to use the same weights matrix for bond 2 and bond 3 of your chain as for interior bonds, you may perform step 4 above multiple times, specifying a slightly different sequence of groups in each case, as indicated above. In the case of simple chains such as polyethylene, it is also appropriate to use the same matrix for the secondtolast bond of the chain, specified as:
Group 1 = methylene
Group 2 = methylene
Group 3 = methylene
Group 4 = methylene
Group 5 = methyl
To enter properties for the first or last bond of a chain, you specify pairs of groups such as methyl, methylene or methylene, methyl as appropriate.
The convention shown above is not the only possible one for the first weights matrix in a chain. In particular, the form of the bond 1 matrix may vary depending on the structure of the matrix associated with the second bond of the chain.
Strictly speaking, the weights matrix for bond 2 should include contributions only from firstorder interactions (see the Statistical weights section). Because of this, and because the state of the first bond is undefined, the weights for bond 2 can be placed in a diagonal matrix, instead of a matrix whose first row represents the first order interactions for this bond. In the case of a diagonal bond 2 matrix, the bond 1 matrix must have the form:
Another possibility is that the reference state (e.g., the trans state) for bond 2 is not the state represented by the first row of its weights matrix. When statistical weights are calculated automatically using the commands in the Stat_Weights pulldown, this is a distinct possibility. The order of rows (and also columns) in this case, is from the most negative to most positive value of the torsion angle. (Thus, for polyethylene, the order of the states would be gaucheminus, trans, gauchepluscorresponding to the angles 120° 0°, +120°). If such a convention is used, and the bond 2 matrix is not of diagonal form, then you must either:
The final bond of the chain has the following sequence:
Group 1 = methylene
Group 2 = methylene
Group 3 = methylene
Group 4 = methyl
Group 5 = NONE
and also has a special weights matrix that is a _{n1} X 1 column vector of the form:
To use the RIS_Monte_Carlo/Range command, select the function for which you wish to set the argument range. Then enter values for Min Argument, Max Argument, and Number Of Bins. The parameters of Min Argument and Max Argument are self explanatory. The Number Of Bins parameter determines the number of points between the minimum and maximum argument at which the function is to be calculated.
You may use the RIS_Monte_Carlo/Range command for any number of functions before invoking the RIS_Monte_Carlo/MC_Run command. If you choose not to use the Range command for a function, reasonable default values for Min Argument, Max Argument, and Number Of Bins are used when you calculate the function.
Each scattering center that you identify has a Scatterer Number associated with it. For the purpose of calculating partial scattering functions, however, it is the Scatterer Name that is relevant. You may have several different scattering centers that you select identified by the same Scatterer Name. However, no more than four distinct Scatterer Names are allowed.
To identify a scattering center, you must first identify the key backbone atom associated with it. The key backbone atom of a group is the atom in the chain's backbone with which the group is identified. In the case of a simple group such as methylene, this is an unambiguous selection. In the case of a more complicated group, such as pphenylene, there is a certain arbitrariness in identifying the key atom. In such a case, the key atom is defined as the tailmost backbone atom in the group. Because of the head/tail arbitrariness in a polymer chain, you may have to select first the atom at one end of a group and then that at the other end in order to identify the key atom. (You are prompted with an error message if a Backbone Atom you select is not a key atom.)
Once a key atom is selected, you may simply let the key atom be the scattering center, or you may select up to three additional atoms to establish the position in space of the scattering center. The average position of these auxiliary atoms (Scatter Atom 1, 2, and 3) gives the scattering center position. In performing the Monte Carlo calculations, this position is assumed to be invariant in the coordinate frame associated with the bond connecting the key atom with its immediately headward key backbone atom. (See Flory 1989, page 20, for the definition of this coordinate frame). When you have identified a scatterer, all groups with the same key as that of the key backbone atom are identified with scatterers of the same type.
The specific steps in identifying scatterers are as follows:
Select the Read_Configs option and enter the name of the Configuration File you wish to read. Select Execute. The file is read, and the Display_Config option is automatically selected. You may then repeatedly select Execute to step through the configurations, or enter any specific Config Number that you wish to see.
When you wish to calculate many different functions for a large chain, it is best to calculate the order n and order n^{2} functions separately. The order n functions typically require many more configurations than do those of order n^{2}. (Note: for S(k), an approximate order n calculation method is available, and may be selected using the RIS_Monte_Carlo/Range command. For chains containing more than about 100 bonds, this method produces results of comparable precision to the order n^{2} method in a much shorter time).
A chain of only one bond results in an error message. This is a trivial case, of course. A chain of only two bonds is likely not to have statistical weights assigned, and thus may generate an error message. (A special weights record must be entered for the twobond case, with the first and last groups both being NONE. Since the twobond case is normally of little interest, this has not been done for the system statistical weights library).
Note that the above restrictions apply only to RIS and RIS MC calculations. They do not apply to RMMC calculations, which are discussed next.
To calculate a bond pair distribution function, use Subset/Template to define a subset of ntuples containing 5 to 8 atoms each. The first 4 atoms of each ntuple define the first dihedral angle of a bond pair; the last 4 atoms define the second. Execute the RIS_Metrop_MC_Run command with the Dihedral_Pair_Dist parameter toggled on. The RMMC program will calculate both a priori and conditional probability distributions for the bond pair.
Then, from the Open Tutorial window, select Polymer Modeling and Property Prediction tutorials, then select RIS Module Tutorials, and choose from the list of available lessons:
Lesson 1: Calculating Properties of Isotactic and Syndiotactic Polystyrenes
Lesson 2: Using RIS Metropolis Monte Carlo Simulation
Lesson 3: Calculating RIS properties for a new polymer is not included as an online tutorial. For this tutorial, follow the stepbystep instructions given below.
You can access the Open Tutorial window at any time by clicking the Open File button in the lower left corner of the Pilot window.
For a more complete description of Pilot and its use, click the onscreen help button in the Pilot interface or refer to the Introduction in the Insight II manual.
> mkdir ris_tut > cd ris_tut
In order to carry out RIS calculations, all the groups in the polymer must be present in a RIS groups database and statistical weights for all the bond pairs must be present in a RIS statistical weights database. It is suggested that you use local copies of the RIS groups and weights databases rather than modifying the centrally located files.
It should be emphasized that the statistical weights you calculate here are based on minimizations in vacuum. Thus, any effects due to surrounding chains or solvent in a bulk system are neglected. Although weights computed in this manner may correctly predict trends within families of polymers, they should be validated and fine tuned by comparison with experimental data whenever possible. This optimization has been done for virtually all statistical weights reported in the literature.
The topics covered in this tutorial are:
RIS module
 

RIS_Compute

Database

Stat_Weights


Files  Properties  Bond_Pairs 

C_Run  Weights  Define_Torsions 

Define_Segment  Dihdrl_Dihdrl_Run 
 
DihdrlDihdrlCntour 
 
Analyze_Map 
 
Compute 
 
S_Run  Get  Edit 

Substitute  Contour 
 
Group_Database 

Polymerizer module
 

Homopolymer

Forcefield



Polymerize  Potentials 


Perform the following procedure in the UNIX environment before started Insight II.
This creates a statwt_tut subdirectory and copies to it the files you need for this lesson.
2. Set the Environmental Variable RIS_FILES
Set the environment variable for the RIS files to point to your statwt_tut directory. This tells the RIS program where to look for the RIS database files. You can also change the environment variable within Insight using the Session/Env_var command. If you always want to use local versions of these RIS database files you can place the setenv RIS_FILES command in your .cshrc file.
To set the required environment variable, and verify that it points to your statwt_tut directory, enter:
> setenv RIS_FILES $cwd
> printenv RIS_FILES
From the statwt_tut directory, start an Insight II session by entering insightII at the UNIX operating system prompt.
4. Sequence the Polymer and Display Unknown Groups
Click the Accelrys icon and select the RIS module. You are prompted to Fix the potentials in the Forcefield/Potentials command. Select Fix for Potential Action and Charge Action, then Execute.
The following message appears in the information area:
Starting S_Run background job on <hostname> as job <job number>After a few moments the Sequencer background job completes and the following message appears in the information area:
S_Run job terminated with diagnostics: There are unknown groups in the chain.Next you display the unknown groups in the polymer.
Make sure that the end with the Tail label is on the left.
In the poly(vinylidene chloride) molecule you have built there are six labeled atoms, i.e., one unknown group which occurs five times, plus the tail atom, which is also labeled.
5. Add Unknown Groups to the Groups Database
Here, you add dichloromethylene to the groups data base.
The information area contains the following messages after the Add is performed:
Starting S_Run background job on <hostname> as job <job number>Notice that because dichloromethylene is not pseudochiral, choosing the Tailward Atom to be on the right side of the (CCl_{2}) group would not make any difference in the group definition.
S_Run job completed successfully.
You next resequence the polymer into groups.
Select the Sequencer/S_Run command. Select Execute.
After a few moments the Sequencer background job is completed and the following message appears in the information area:
S_Run job completed successfully.6. Verify the Absence of Known Groups
The information area contains the following message:
There are no unknown groups.Notice that there are no labeled atoms on the polymer.
7. Employ Local User Files Instead of Default System Files
User Weights File: user.statwt
Select the RIS_Compute/Files command.
Stat Weights File: User (default is System)
This command allows you to employ user.statwt as the statistical weights database instead of the default file $BIOSYM/data/polymer/ris/system.statwt, and user.prop as the properties file instead of $BIOSYM/data/polymer/ris/system.prop. Note that because you defined the RIS_FILES environment variable, there are no system files shown. If, after finishing this tutorial, you wish to access the system files, open a new UNIX window.
The local file user.statwt is empty at this point. You will enter the statistical weights for poly(vinylidene chloride) into this file after they are calculated.
The local file user.prop is also empty. You will add the bond lengths associated with poly(vinylidene chloride) into this file.
8. Add Properties for the New Groups to the Properties Database
Group 1: dichloromethylene (select from the list
Select the Database/Properties command.
Action: Add
Next, specify the same bond length by choosing methyl for Group 1 and dichloromethylene for Group 2.
In general, if you enter properties for AB, where either A or B is a new group, you should also enter properties for BA.
Repeat the same process for the groups in the following table:
Group 1  Group 2  Bond Length 

methylene  dichloromethylene  1.53 
dichloromethylene  methylene  1.53 
hydrogen  dichloromethylene  1.10 
dichloromethylene  hydrogen  1.10 
9. Identify and Extract Bond Pairs for Which Statistical Weights
Are Not Known
Select the Stat_Weights/Bond_Pairs command. Select Locate_Unknown for Bond Pair Action. The molecule name (POLYDCLE) is automatically entered for the Molecule Name parameter. Select Execute.
The following message appears in the information area:
There are 6 unique unknown bond pairs.
Select Execute to display the first "pair". 
Groups identified are: methyl dichloromethylene methylene
Select Execute again to display the second bond pair. 
Groups identified are: methyl dichloromethylene methylene dichloromethyleneNote that the first bond "pair" is identified by three consecutive backbone atoms, and the second "pair" is identified by four such atoms.
It should be noted that a bond pair is defined by a sequence of five groups. In such a sequence, the bond connecting the backbone atom of group 2 to that of group 3 is the first bond, whereas the bond connecting the backbone atom of group 3 to that of group 4 is termed the second bond.
Therefore, strictly speaking, the first two sequences of groups located and displayed by the Stat_Weights/Bond_Pairs command do not define bond pairs. The Bond_Pairs command identifies these two sequences as "unknown", because the statistical weights database has to contain a matrix corresponding to each of these sequences before you can calculate conformation dependent properties for the polymer in question using the RIS module.
The matrix corresponding to the first sequence (CH_{3}CCl_{2}CH_{2}) is a row vector (given by Eqn (43) in Flory 1974) and does not require any calculations. The specification of this matrix will be discussed later in this lesson.
The second sequence (CH3CCl2CH2CCl2) identifies the second bond (the first internal bond) on the backbone of the polymer chain. The statistical weight matrix (denoted by U_{2} on pp.6768 of Flory 1989) for this bond can be calculated automatically by the Stat_Weights tools. This matrix can be approximated, however, by using the weight matrix of the internal bond pair defined by the sequence CCl2CHCCl2CH2CCl2 (see pp.6768 of Flory 1989).
Select Execute again to display the third bond pair.
The information area contains the following message:
Groups identified are: methyl dichloromethylene methylene dichloromethylene methyleneThe backbone atoms of these groups are labeled.
These five consecutive groups define a bond pair for which the statistical weights can be estimated using the procedure described below. The statistical weight matrix for this bond pair, however, is the same as that for the fifth bond pair defined by the groups:
methylene dichloromethylene methylene dichloromethylene methyleneA separate calculation for the third bond pair, therefore, is not necessary. The results that will be obtained for the fifth bond pair will be employed for this pair also. (To see the basis of this usage, you may note that both sequences contain the same bond pair: CCl2CH2 and CH2CCl2. Furthermore, the neighboring backbone atoms are carbon atoms, either in the form of CH3 or CH_{2}C, which are essentially the same for our purposes).
Select Execute again to display the fourth bond pair.
The information area shows the following:
Groups identified are:
dichloromethylene methylene dichloromethylene methylene dichloromethylene.
Now, set Bond Pair Action to Extract_Unknown, Type DCL1 for the Fragment Name, and select Execute. 
Move this segment (DCL1) to the upper left side of your screen using the mouse.
Bond Pair Action: Display_Unknown
The information area shows the following:
Groups identified are: methylene dichloromethylene methylene dichloromethylene methylene
Set Bond Pair Action to Extract_Unknown, type DCL2 for the Fragment Name, and select Execute. 
The information area shows the following:
Groups identified are: methylene dichloromethylene methylene dichloromethylene hydrogenThis sequence of groups, while not equivalent to the fifth sequence, contains the same bond pair: CCl2CH2 and CH2CCl2. Furthermore, it is on the "right" end of the chain and occurs only once along the chain. Therefore, its statistical weight matrix is approximated by that of the fifth sequence, the error introduced being negligible for long chains for which end effects are not significant.
Thus, there are six unknown bond pairs (i.e., six sequences of groups) for which you have to enter the statistical weight matrices into the statistical weights database before you can calculate conformation dependent properties for this "new" polymer using the RIS module.
These six sequences of groups are:
The calculations carried out using DCL1 will give the matrix for sequence 4.
The matrix for sequence 1 is a row matrix, which can be determined without any calculations (these will be described later).
The matrix for sequence 2 will be approximated using that of sequence 4.
At this point, delete the original polymer chain using the Object/Delete command.
10. Calculate Statistical Weights for the Internal Bond Pairs
These 4 consecutive backbone atoms define the first rotational angle and are shown in the illustration below:
Now type phi2 for Torsion Name and pick the 4 backbone atoms starting from the second backbone atom from the left end of the segment: 
This sequence of atoms defines the second torsion angle in your molecule.
The following message is displayed at the information area:
Starting Discover minimization background job on <hostname> as job <job number>
Repeat the above commands (starting with the Stat_Weights/Define_Torsions command, and ending with the Stat_Weights/Dihdrl_Dihdrl_Run command) for the segment DCL2. 
Starting Discover minimization background job on <hostname> as job <job number>Even with the coarse grid used by default (i.e., 36°), the Discover calculations take about ten minutes on a 20 MHz 4D/25 SGI workstation to complete. To obtain maps with enough detail that "good" estimates for the statistical weights can be obtained using them generally requires a finer grid and several hours of CPU time.
The main purpose of this lesson is to familiarize you with the procedure of constructing contour maps of energy, analyzing these maps and calculating statistical weights, which is the same whether the grid you use is fine or coarse.
Because of the consequent large increase in CPU time and disk space requirements, the use of an angle increment smaller than 10° is usually not warranted. Also note that the angle increment you specify should divide 360.0 evenly.
When the first Discover job is complete, the following message is displayed at the information area:
Job <job number> (Dihdrl_Dihdrl_Run DCL1) completed; status 0You now proceed to calculate the statistical weights for DCL1. (You later perform the same step for DCL2.)
Discover job has successfully completed.
After the first Discover job is complete, select the Stat_Weights/DihdrlDihdrlCntour command.
Contour Step: Construct_Graph
A 3D graph showing the total energy of the molecule as a function of two dihedral angles is constructed and displayed.
Note: You should enter an energy value that is slightly greater than the minimum energy that you can read from the graph. For example, if the axis minimum is labeled 4.4 then you should enter 4.45 as the minimum value. You need to do this because Graph labels are rounded off, and if you enter a minimum energy less than the true minimum on the graph, you will get an error message.
Select the Stat_Weights/Analyze_Energy command. Select Execute.
This command analyzes the given map to find the minima on it. A set of default regions is generated and displayed. Each such region represents a state in the RIS approximation.
This command computes the average angles, partition function, and average energy for each region on the map. The average angles are displayed (using yellow triangular marks) upon the completion of this command.
Select the Combine_Angles option for Compute Step, and Execute.
The statistical weights for the bond pair are calculated and the results are displayed in a table named RIS_DCL1. The weights are also written to a file named DCL1.weights.
11. Add Statistical Weights to the Database
The statistical weights for the two internal bond pairs of poly(vinylidene chloride) are in the tables named RIS_DCL1 and RIS_DCL2. The next step is to add these weights to your local database file user.statwt.
Table Name: RIS_DCL1 (select from the valueaid)
User Weights File: user.statwt
Choose the following as Group 1 through Group 5:
Group 1: dichloromethylene
Select the Database/Weights command.
Nu0: 3
As an approximation, you will use the same matrix (appropriate for interior bonds) for the second bond from the tail end of a chain. The matrix RIS_DCL1 contains contributions from both first and second order interactions (see the Statistical weights section), and in principle the matrix for the second bond should contain only first order interactions. However, the error introduced by this approximation becomes vanishingly small for long chains.
While still in the Database/Weights command, change the group names to the following:
Group 1: NONE
Thus, the terminal group sequence CH_{3}CCl_{2}CH_{2}CCl_{2} is assigned the same statistical weights as the internal group sequence CCl_{2}CH_{2}CCl_{2}CH_{2}CCl_{2}.
Next, store the matrix RIS_DCL2.
Lastly, you must add the weights for the first bond of this polymer, because the results of the Stat_Weights/Compute command are not in the standard Flory form (see the Terminal and nearterminal bonds section). ^{4}
Remain in the Database/Weights command.
Weights_Action: Create_Wts_Table
Nu0: 1
Nu1: 3
(these parameters give the dimensions of the matrix)
Type FIRST for Table Name, and select Execute.
This indicates that the dihedral angles for the states of the first bond are all zero. (The state of the first bond is actually undefined, but by convention it is assumed to be always trans. See Flory 1974, p.386.)
These are the prefactor (A1k) and energy (EPS1k) values for each state. When complete, the matrix should appear as given below:
FIRST  1  2  3  4 
1  PHIk  0  0  0 
2  A1k, EPS1k  0, 0  1, 0  0, 0 
If your matrix looks different from this, use the mouse to set the focus to the appropriate cell, and retype its value. When the matrix is satisfactory, set Weights_Action to Add. 
Enter the following for the group names, and then select Execute:
Group 1: NONE Group 2: NONE Group 3: methyl Group 4: dichloromethylene Group 5: methylene

12. Verify Presence of Statistical Weights for Each Bond Pair in the Polymer
If you have deleted your original polymer, rebuild a chain of poly(vinylidene chloride) using five repeat units.
Next, select the Stat_Weights/Bond_Pairs command. Select Locate_Unknown for Bond Pair Action, enter the name (POLYDCLE) of your polymer, and select Execute.
The following message is displayed at the information window:
There are no unknown bond pairs13. Calculate Conformation Dependent Properties
Now that you have entered all the necessary properties of poly(vinylidene chloride) in your local database, you will carry out a simple RIS calculation to obtain the characteristic ratio as a function of number of bonds using the statistical weights you have calculated.
To perform RIS calculations on chains of variable length, you must first use the RIS_Compute/Define_Segment command to define head, tail, and repeating segments for your chain. In this lesson, you will do this for the poly(vinylidene chloride) chain you have constructed.
This causes the key atom of every group in the chain backbone to be labeled with its atom name.
When this is done, the command executes, and the Segment Type is automatically set to Repeating.
Select the RIS_Compute/C_Run command.
Run Name: PVDC
When the RIS calculation is complete, the following message is displayed in the information window:
RIS job completed successfully
Select the Graph/Get command. File Name: PVDC.ristbl 
^{1}
For example, in the case of pphenylene, it is as if an imaginary atom lies at the center of the ring. The virtual bonds connect this imaginary atom to neighboring groups.
^{2} The superscript t on E^{t} stands for total, indicating that this is the total energy of the segment.
^{3} Note that for the common 3state case, the order of the rows and columns used here is g^{}, t, g^{+}; as opposed to the common arrangement t, g^{+}, g^{}.
^{4}
Each row of the weights matrix corresponds to a distinct phi1 value, and each column corresponds to a distinct phi2 value. The rows and columns of a matrix generated by the Stat_Weights tools are put in ascending order of phi1 and phi2 values, respectively. Thus, the phi1 value corresponding to the ith row is smaller than that corresponding to the (i + 1)th row.
Hence, for symmetric vinyl chains, the order is g, t, g+ instead of t, g+, g. This approach permits consistent handling of all linear polymers: not all chains can be handled using the familiar 3state (t,g+,g) model.
For example, the trans state (i.e., phi = 0) may not be one of the low energy states, and therefore may not be used as a rotational state in the RIS model.
Note also that you should not mix the weight matrices generated by the Stat_Weights tools by those in the system database. That is, if the weight matrices of some of the bond pairs are available in the system database, and you calculate the unknown weight matrices using the Stat_Weights tools, you should not mix these matrices. The correct approach for a such a polymer is to calculate all the weight matrices using the Stat_Weights tools and put these matrices into a new (user) database as described in this lesson.
Last updated April 20, 1998 at 10:02AM PDT. Copyright © 1997, Accelrys Inc. All rights reserved.