*[Chapter 2.1*] 006 .ls 2c 006 .tg NI 005 .ul 1 057 Chapter 2 : Methods for Calculating Interaction Energies. 005 .tc : 007 .ta 56r 041 :"We use the classical theory on Mondays, 040 :Wednesdays and Fridays, and the quantum 034 :theory on Tuesdays, Thursdays and 012 :Saturdays." 018 :Sir William Bragg 003 .tc 006 .ls 3c 006 .tg NA 064 Some of the calculations we shall perform require decisions 071 to be made as to the stability of given structures and conformations of 053 molecular systems. Largely, these involve measuring 064 the energy of the systems, but, because of the assumptions which 061 have to be made in the calculations, great care must be taken 028 in interpreting the results. 071 A complete theoretical determination of the equilibrium properties 053 of a system would require evaluation of free energies 060 associated with the system, taking into account such factors 067 as solvation, other species present in solution, and so on. It is 063 doubtful whether it is possible to carry out this type of study 067 at present on any system of practical interest; even assuming that 044 the theoretical tools are capable of dealing 058 with the problem, the computation required is prohibitive. 065 On systems of biological importance, such as the DNA complexes of 059 interest here, which involve large molecules interacting in 048 ionic media of dubious composition, the problems 057 associated with a complete determination of free energies 053 become insurmountable. Instead, it has become usual 057 to study the interactions between molecules *Iin vacuo*N, 052 ignoring all solvent effects, and to assume that the 059 solvated systems. In some simple cases (see, for example, 070 Weintraub*S1*N), it has been found that the same optimum conformations 054 of molecules were predicted whether or not the solvent 069 environment was included in the calculation, but the systems on which 068 these findings were based possessed no net charge. This electrical 054 neutrality has, in fact, characterised practically all 063 systems studied using the methods of theoretical conformational 055 analysis; however, those systems of interest to us are 058 not electrically neutral and the effects of introducing an 039 overall charge are not well understood. 067 It is necessary to make further assumptions. As noted in the 077 preceeding paragraph, it is the free energy of a molecular system which is of 058 chief interest. This free energy can be written as a sum 042 of an internal energy and an entropy term: 006 .sp 1c 005 .ce 1 016 *DA *N= *DU - TS 008 *N.sp 1c 060 Although methods are available for evaluating conformational 068 entropy terms (see Hopfinger*S2*N), these are not very satisfactory, 062 because it is difficult to sample an adequate cross-section of 065 the conformational space available to systems of interest. This 065 leaves only the internal energy, *DU*N, to be calculated, and all 056 our work, in common with most previous studies, is based 057 on considerations of energy terms which are related to an 029 *Iin vacuo *Ninternal energy. 071 Determining the *Iin vacuo *Ninternal energy of a molecular system 060 could, in principle, be done by solving the Schrodinger wave 069 equation*S3 *Nin a form appropriate to the system in question. Like 060 much of the mathematics which arises from natural phenomena, 065 the states available to a molecular system are described in terms 067 of wave-functions which are eigenvectors of a differential operator 075 based on the Laplacian *SV*N*t2*l, and the energies of the states are given 060 by the corresponding eigenvalues. As expected, though, the 063 equation hides the complexity of the real world, and it has not 062 proved possible to obtain exact solutions to this equation for 044 systems more complex than the hydrogen atom. 062 Numerical solutions can be obtained to any degree of accuracy, 047 although at considerable computational expense. 070 More useful for the study of interactions between large molecules 073 are methods which do not attempt *Iab initio *Ndeterminations of chemical 064 properties, but which instead use fairly drastic approximations. 069 These methods can give reliable results using much less computer-time 067 than the *Iab initio *Nmethods, partly because errors introduced by 070 one approximation can cancel out those produced by another, and partly 062 because quantities which would be calculated in an *Iab initio 061 *Ncalculation are parameterised from experiment. This field 055 of quantum chemistry is full of acronyms describing the 064 approximations being used; two methods of particular importance 076 in quantum pharmacology are CNDO/2*S4 *N(C_omplete N_eglect of D_ifferential 045 O_verlap), in which many integrals are either 071 omitted or parameterised from experiment, and PCILO*S5 *N(P_erturbative 067 C_onfiguration I_nteraction using L_ocalised O_rbitals), which uses 068 the main approximations of CNDO/2, but replaces the normal iterative 066 matrix diagonalisation process by a faster perturbation treatment. 059 This makes PCILO particularly suitable for the rapid survey 045 important compounds, but both methods produce 034 results which are fairly reliable. 069 The interested reader will find an introduction to quantum mechanical 055 methods in the books by Richards*S6 *Nand Murrell*S7*N. 064 An alternative is to use so-called "semi-empirical" methods 061 to determine properties of molecular systems. These methods 058 are despised by some because of the absence of theoretical 062 justification, whilst being over-interpreted by others. Both 068 viewpoints are extreme; when alternative methods cannot be applied, 064 semi-empirical methods can be quick and reliable. Care must be 028 taken, though, in their use. 074 The total internal energy of a collection of molecules may be written 071 as the energy of the component molecules isolated from one another plus 028 a set of perturbation terms: 005 .ce 1 006 .sp 1c 091 *GE *N= *I*bm*(*S*lRT*I*bo*)l *G*lE*D? *N+ *I*bn*(=*S*lRT*)*I*b1,*SI *G*le*(*I*dp*tn*)*dert 010 *N*l.sp 2c 057 and the interaction energy between the molecules is equal 057 to the sum of the various order perturbation terms. The 059 proponents of the use of such quantities as "van der Waals" 060 energy, "hydrogen-bonding" energy and so on, assume that the 058 perturbation terms can be represented to a suitable degree 064 of accuracy by empirical formulae. Sometimes it is possible to 055 show the theoretical rationale for particular formulae, 064 and then we call these expressions "semi-empirical" to show this 066 connection with theory. An excellent review of the justification 062 behind this method of approximation is given by Claverie*S8*N, 013 who makes the 057 pertinent point that simple empirical expressions, having 062 or better than those given by an *Ia priori *Naccurate method. 064 However, it is worth remembering that semi-empirical techniques, 051 by definition, are parameterised using experimental 068 results, and it can become very difficult to determine the extent to 060 which a result based on such techniques is truly predictive; 054 much may be just a direct consequence of the parameter 012 values used. 064 Whilst a quantum mechanical method such as PCILO would have 067 been applicable to some of the problems investigated in this thesis 066 (although not to all, because many of the systems studied were too 064 large even for PCILO), it would have been difficult to interface 040 the method to the numerical optimisation 068 algorithms which would provide for automatic refinement of molecular 067 conformation. Because of this, we used instead the semi-empirical 009 approach. 059 We accepted that the effect of the perturbation terms could 051 be written as a set of expressions, each defining a 005 .ne 3 031 particular type of interaction: 005 .ce 1 006 .sp 1c 144 *GE*I*dint *N*l= *I*bn*(=*S*lRT*)*I*b1,*SI *G*le*(*I*dp*tn*)*dert *N*l= *GE*I*del *N*l+ *GE*I*dpol *N*l+ *GE*I*ddisp *N*l+ *GE*I*drep *N*l+ .... 006 .sp 2c 067 where *GE*I*del *N*lis the electrostatic energy arising from static 069 charge distributions on molecules, *GE*I*dpol *N*lis the polarisation 054 energy, or reaction of a fluctuating electric field to 070 a static field, *GE*I*ddisp *N*lis the dispersion energy, the reaction 071 of the fluctuating field to itself, and *GE*I*drep *N*lis the repulsion 064 energy arising from overlap of the electron clouds of non-bonded 049 atoms. The remainder of this chapter is devoted 043 to a description of these individual terms. 002 *? *[Chapter 2.2*] 006 .tg NI 005 .ul 1 048 2.1 : Calculation of Electrostatic Interactions. 006 .tg NA 066 In any molecule, the electrons will be distributed throughout 066 space in such a way as to concentrate negative charge into certain 070 regions whilst removing it from others. In the overwhelming majority 047 of molecules, some atoms possess, on average, a 058 net positive charge, whilst others possess a net negative 067 charge. It has long been the practice to use this distribution of 073 positive and negative charges to calculate part of the interaction energy 040 between molecules, using the expression: 006 .sp 1c 005 .ce 1 086 *GE*I*del *N*l= *(*SRT*I*bi*) *(*S*lRT*I*bj*) *N*lB.*Iq*di*N*l.*Iq*dj *N*l/ *Se*Ir*dij 010 *N*l.sp 2c 078 (where B = 1368 kJ.A.mol*t-1*l, *Se *Nis the dielectric constant of the medium 082 separating atoms *Ii *Nand *Ij *Na distance *Ir*dij *N*lapart) by assigning static 088 charges *Iq*di*N*l, *Iq*dj *N*lto each atom within the molecules, situated at the atoms' 007 nuclei. 067 Determination of the static charge distribution in a molecule, 062 however, requires some form of quantum mechanical calculation. 024 Fortunately, it is found 061 that the static charge distribution in a molecule is *Ifairly 059 *Nindependent of the molecule's environment; indeed, it is 061 possible to determine the distribution in a large molecule by 060 splitting it up into *Ichemically sensible *Nfragments*S9*N. 077 Hoyland*S10 *Nhas reviewed the application of common approximations used 067 in quantum mechanics to the determination of quantities of chemical 064 interest. He found that the CNDO/2 approximation, developed by 059 Pople and Segal*S4*N, produced good estimates of quantities 056 we based most of our calculations on this method. Some 076 early calculations were based on the Pariser-Parr-Pople*S11 *Napproximation, 068 but this method, on investigation, proved less reliable than CNDO/2. 066 We compared values for residual atomic charges obtained using 044 these two approximations with experimentally 072 determined dipole moments and with *S13*NC nmr shifts. The theoretical 064 relationship between residual atomic charge and molecular dipole 065 moment is fairly clear, but is of little value in determining the 062 accuracy of the atomic charges, since there is only one figure 056 to correlate with all the charges on the molecule. The 066 relationship between atomic charge and nmr shift should be of more 064 value, since nmr shifts are available for most of the atoms in a 061 molecule, but unfortunately, the relationship is not perfect; 072 although the resonant frequency of a nucleus depends on the shielding of 063 that nucleus from an applied magnetic field and thus depends on 059 the electron density around that nucleus, other factors are 009 involved. 067 Nonetheless, an approximate relationship between nmr shift and 005 .ne 3 026 atomic charge of the form: 006 .sp 1c 005 .ce 1 034 *Sd*I*di *N*l= *Sa*Iq*di *N*l+ *Sb 008 *N.sp 2c 071 can be written, where *Sd*I*di *N*lis the shift of resonance of nucleus 088 *Ii *Nfrom *Sb *Nand *Iq*di *N*lis the partial charge on that atom. (See, for example, 025 Karplus and Pople*S12*N.) 072 Investigations of this relationship using *S13*NC nmr data have produced 054 values for *Sa *Nof between -60 and -200 with aromatic 014 systems*S13*N. 031 The *S13*NC nmr spectra of 059 acridine, proflavine and 9-aminoacridine, in protonated and 068 Roche Products Ltd. Assignment of certain peaks was possible using 063 standard techniques and, by using these values along with those 059 published for the quinoline system*S14*N, we determined the 071 correlation between these values and the CNDO/2 and PPP atomic charges. 069 The results are given below in figure 2.1; it can be seen that there 063 is a much closer correlation with the CNDO/2-determined charges 037 than with the PPP-determined charges. 053 Figure 2.2 shows the correlation between experimental 067 dipole moments, obtained from McClellan*S15*N, and those determined 041 by CNDO/2 calculations. No particularly 062 bad discrepancies are apparent, such as are found with simpler 037 quantum mechanical approximations and 055 we therefore confirm previous authors' conclusions that 065 the CNDO/2 technique is moderately reliable in its predictions of 051 molecular properties associated with atomic charge. 035 Given a set of atomic charges, 039 we can evaluate an electrostatic energy 069 for a molecular system from the preceeding expression, providing that 056 the value of the dielectric constant *Se *Nis specified. 021 Previous workers have 070 commonly used either a constant value to describe dielectric behaviour 073 or, more recently, a function which increases as the distance between the 078 interacting charges increases*S2*N. The rationale for this latter choice can 074 readily be understood; since the dielectric constant describes the way in 073 which intervening matter changes the interaction between charges, it must 069 become small as the charges approach one another and as the amount of 076 intervening matter becomes small. The limiting value as the charges become 073 indefinitely close is unity, whereas at large separations, the value will 076 Previous workers have found that they could most accurately predict the 075 behaviour of biological macromolecules using a dielectric constant function 081 which increased to between 2 and 4 at large interatomic separations*S16*N. This 072 is interesting, since it shows that, at least for the systems studied by 071 these workers, the solvent environment around such macromolecules, with 071 bulk dielectric constant of about 80, does not greatly affect the ionic 024 energy of the molecules. 057 We took *Se*N=1 at separations less than 4A, tending to 3 031 at separations greater than 7A. 078 One point remains concerning the manner of variation of such a dielectric 009 function. 032 Initially, we used the piecewise 069 linear function as shown in figure 2.3 and described, for example, by 076 Hopfinger*S2*N. This function can be evaluated quickly, but unfortunately, 050 since its gradient is discontinuous, its use slows 032 the rate with which optimisation 048 algorithms locate minimum energy configurations. 071 Consequently, we adopted the second function shown in figure 2.3, which 070 varies in a similar fashion to the piecewise linear function but which 033 possesses continuous derivatives. 002 *? *[Chapter 2.3*] 006 .tg NI 005 .ul 1 038 2.2 : Dispersion and Overlap Energies. 006 .tg NA 068 We turn now to the interaction terms which would conventionally 065 be termed "van der Waals" interactions. These correspond to the 075 combination of favourable terms arising from the alignment of instantaneous 070 dipoles on atoms ("dispersion" energies) and the unfavourable energies 068 produced when the electron clouds on non-bonded atoms overlap to any 062 to calculate an overall energy arising from these two effects: 006 .in 10 006 .sp 1c 006 .ti -5 044 1.ZZZThe Lennard-Jones 6-12 potential*S17*N: 003 .br 123 *GE*I*dvdw *N*l= *(*SRT*I*bi*) *(*S*lRT*I*bj*) *lA*dij*N*l/*Ir*(*S1*I*di*)*(*S*l2*I*dj*) *N*l- *IC*dij*N*l/*Ir*(*S6*I*di*)j 010 *N*l.sp 1c 006 .ti -5 036 2.ZZZThe Buckingham potential*S18*N: 003 .br 125 *GE*I*dvdw *N*l= *(*SRT*I*bi*) *(*S*lRT*I*bj*) *lA*('*di*)j *N*lexp(-*IB*dij*l.r*dij*N*l) - *IC*('*di*)j*N*l/*Ir*(*S6*I*di*)j 010 *N*l.sp 1c 005 .ti 5 065 where *Ir*dij *N*lis the separation beteen atoms *Ii *Nand *Ij*N, 006 .sp 1c 005 .in 0 046 although other expressions are available*S8*N. 077 The *Ir*t-6 *N*lattractive term, which represents dispersion energy, can 065 be derived rigorously from quantum mechanics for two isolated and 070 spherically symmetric systems (although there are higher-order terms), 036 but its validity when applied to the 070 interaction of individual atoms within molecules is more questionable. 081 The constants *IC *Nand *IC' *Ncan be expressed in terms of atom polarisabilities 060 using a modified form of the Slater-Kirkwood*S19 *Nequation. 062 The form of the repulsive term cannot be derived so rigorously 070 (although there is more justification theoretically for an exponential 081 term*S20 *Nthan an *Ir*t-12 *N*lterm), and the use of either merely expresses the 066 fact that there is a very sharp rise in energy when atoms approach 024 one another too closely. 071 However, as the interatomic separation tends to zero, the Lennard-Jones 065 potential increases indefinitely, whilst the Buckingham potential 066 reaches a maximum and then decreases to -*SI *Nat zero separation. 045 (This point has been noted by Claverie*S8*N.) 062 a molecular structure automatically on the basis of its energy 035 involve generating fairly arbitrary 035 configurations of atoms, it is easy 055 to produce totally unreasonable configurations in which 065 atoms had become almost superimposed. If a Buckingham potential 060 was used to calculate the energy of such a configuration, an 067 apparently favourable value would result. (We do not believe that 067 nuclear fusion plays an important role in stabilising biopolymers.) 065 Those configurations which are important in determining molecular 070 properties of interest here are those which are chemically reasonable, 072 and we feel that it is relatively unimportant if unstable configurations 066 are incorrectly described, provided they are shown to be unstable. 026 We therefore used a wholly 072 Lennard-Jones potential and obtained values for the constants *IA*N, *IC 022 *Nfrom Scheraga*S21*N. 058 Scheraga's treatment of van der Waals interactions is 055 interesting in that it accepts a fundamental difference 070 between, for example, aromatic and aliphatic carbon atoms. Different 069 values for the constants Aij and Cij were obtained for each of twelve 068 different atom types, corresponding to different environments around 070 these atoms; we give these types in table 2.1. (Computationally, it 061 is straightforward to determine an atom's environment using a 069 graph representation of the bonding in the molecules, as described in 010 *S?*N1.3.) 061 This parameterisation was used with only minor modifications. 035 Because no values were included for 058 phosphorus, this was treated using the sulphur parameters. 055 Dispersion interactions with phosphorus were relatively 068 workers*S22 *Nhad found that sulphur interactions could be predicted 059 well using parameter values obtained in a similar potential 057 function for argon, and it seemed unlikely that any gross 059 errors would be incurred by treating phosphorus as sulphur. 060 Some discussion of the accuracy of the parameterisation 047 is given in *S?*N5.2. The values used for the 045 constants Aij and Cij are given in table 2.2. 002 *? *[Chapter 2.4*] 006 .tg NI 005 .ul 1 025 2.3 : Other Energy Terms. 006 .tg NA 037 When we expressed intermolecular 062 interaction energy as a sum of perturbation terms, we included 064 a term *GE*I*dpol*N*l. This corresponds to the reaction of the 051 electron cloud to the static charge distribution in 060 a molecular system and is generally called the "polarisation 062 energy". Methods for evaluating this energy are discussed by 064 Claverie*S8*N, and we shall not give them here. Unfortunately, 055 polarisation energies cannot be represented by a sum of 028 atom-atom interaction terms, 048 and they are more difficult to evaluate than the 057 contributions previously discussed. Claverie points out 053 that the polarisation contribution often turns out to 061 be significantly smaller than the others, which is fortunate. 037 Consequently, we did not include 061 polarisation terms in our calculations. It is worth noting, 052 though, that Scheraga*S21 *Ndid not include the term 037 when parameterising the Lennard-Jones 041 potential. It might be argued that some 057 allowance for polarisation contributions was already made 057 in his parameterisation; certainly, it would be wrong to 051 evaluate them again without taking account of this. 059 Three other contributions to the interaction energy of 067 molecular systems were considered. Firstly, we adopted Scheraga's 068 hydrogen-bonding potential*S21*N, since the parameterisation of this 068 resulted from the calculations used for the Lennard-Jones potential. 020 The form of this is: 006 .sp 1c 005 .ce 1 093 *GE*(*I*di*th*)*(*dj*tb*) *N*l= *IAr*(*t-*di*)*(*t1*dj*)*t2 *N*l- *IBr*(*t-*di*)*(*t1*dj*)*t0 010 *N*l.sp 1c 057 with the *Ir*t-6 *N*lattractive term of the Lennard-Jones 067 potential replaced by an *Ir*t-10 *N*lterm to represent the shorter 065 range of hydrogen-bonding interactions. Unlike other potentials 060 used to describe the hydrogen-bond, the function contains no 059 angular dependance. Although there is a marked preference 057 for certain dispositions of atoms in a hydrogen-bond, the 061 effect of the non-bonded interactions between the heavy atoms 055 involved in the bond can be expected to reproduce this. 064 The remaining terms related to bond-bending and stretching. 062 These were needed very rarely; in most of the calculations we 059 performed, bond-lengths and bond-angles were kept constant, 057 because the amount of change which could occur would have 065 marginal effect on the intermolecular interactions. However, in 065 the final stages of refinement, we allowed all atomic coordinates 064 to vary independently, and it was necessary therefore to include 063 constraints on the bonding. We used a very simple Hooke's Law 018 type of potential: 006 .sp 1c 005 .ce 2 075 *GE*(*I*di*tb*)*(*dj*to*)nd *N*l= 1500(*Ir*dij*N*l-*Ir*(*D?*I*di*)j*N*l)*t2 075 *G*lE*(*I*di*tn*)*(*dj*te*)xt *N*l= 900(*Ir*dij*N*l-*Ir*(*D?*I*di*)j*N*l)2 006 .sp 1c 064 high to ensure that bond-lengths and angles remained constant to 060 good accuracy. No attempt was made to give different types 064 of bond different constraining weights, and indeed the constants 023 chosen were rather low. 006 .tg NI 005 .ul 1 014 2.4 : Summary. 006 .tg NA 063 Discussion of the accuracy of the semi-empirical potential 064 functions described in this chapter can be found in the original 064 literature from which the formulations were obtained; a lengthy 059 appraisal of the validity of this type of function has been 066 given by Hopfinger*S2*N. Although we make one or two attempts in 062 later chapters to ensure that the predictions obtained through 064 their use were reasonably accurate, we did not attempt to refine 056 the potentials to give correct results. There were two 059 reasons for this. Firstly, whilst it is quite possible to 057 include extra terms to compensate for the shortcomings of 062 the terms already included, this reduces the predicative value 059 of the functions; a formula which includes (n+1) variables 058 can always be parameterised using 'n' observations to give 061 a correct value for the (n+1)th, but without meaning that the 062 (n+2)th will be reasonable. Secondly, we were concerned with 059 methods for modelling, and not with attempting to calculate 067 rigorously or precisely a given interaction energy. The potential 047 functions described above provided a moderately 056 reliable method for determining the relative stabilities 063 of molecular structures or conformations. They could be coded 049 into a "black-box" within the computer models and 060 would provide an estimate of a system's energy independently 064 to the energy calculations would merely have distracted from the 014 central theme. 064 However, there were limitations to the potential functions. 057 In particular, they could not be expected to predict well 063 interactions between atoms close together in the same molecule, 061 and previous workers have included extra energy terms to deal 059 with bond rotations and so on. Few of the calculations we 061 performed were critically dependent on such terms. When the 060 situation did arise, we attempted to evaluate intramolecular 051 energies directly, using the CNDO/2 approximations. 051 Although residual atomic charges were obtained 030 from the CNDO/2 approximation, 051 we did not repeat calculations already published in 062 the literature, and the residual charges in DNA were therefore 069 taken from Pullman and Pullman*S23*N, and from Olsen and Flory*S24*N. 066 Having described these functions, we turn in the next chapter 065 to a method of modelling which replaces them all by much simpler, 017 quadratic, terms. 002 *? *[Chapter 2 Refs and Tables*] 003 .nj 003 .nf 006 .ls 3c 006 .tg NA 047 T_y_p_e_ D_e_s_c_r_i_p_t_i_o_n_ 006 .sp 1c 043 H*T1 *NAliphatic hydrogen 048 H*T2 *NAmine or amide hydrogen 042 H*T3 *NAromatic Hydrogen 061 H*T4 *NHydroxyl or carboxylic acid hydrogen 041 C*T5 *NAliphatic carbon 059 C*T6 *NCarbonyl or carboxylic acid carbon 040 C*T7 *NAromatic carbon 059 N*T8 *NAmide or NH*(*T3*N*t+*) *lnitrogen 049 N*T9 *NUncharged amine nitrogen 059 O*T11 *NHydroxyl or carboxylic acid oxygen 032 S*T12 *NSulphur 006 .tg NI 005 .ul 3 009 Table 2.1 058 Atom types considered in Scheraga's*S21 *Nparameterisation 031 of the Lennard-Jones potential. 003 .bp 007 .ll 200 006 .ls 2c 006 .tg NA 028 *IAij (kJ mol*t-1 *lA*t12*l) 099 *N1 2 3 4 5 6 7 8 9 10 11 12 010 1 58946 018 2 45825 35234 026 3 59554 46319 60168 034 4 53701 41580 54264 48851 042 5 499748 405853 504022 462620 3787196 050 6 527370 423136 532138 486029 4073985 4384749 058 7 345512 276847 348655 318269 2766910 2930098 1986757 066 8 435567 352323 439363 402618 3402625 3618313 2473130 3062033 074 9 298519 237602 301316 274310 2480736 2594584 1766220 2216640 1567232 082 10 193146 151437 195073 176509 1730397 1766742 1208984 1542743 1065386 711378 090 11 161270 127057 162847 147639 1446792 1470059 1016942 1298585 901945 609463 525141 098 12 254913 204804 257205 235046 2216114 2247752 1596758 2027731 1445545 1016651 885444 1518086 027 *ICij (kJ mol*t-1 *lA*t6*l) 099 *N1 2 3 4 5 6 7 8 9 10 11 12 010 1 190 018 2 190 190 026 3 190 190 190 034 4 190 190 190 190 042 5 525 525 525 525 1549 050 6 774 774 774 774 2212 3204 058 7 525 525 525 525 1549 2211 1549 066 8 512 512 512 512 1531 2171 1531 1518 082 10 509 509 509 509 1538 2170 1538 1529 1606 1542 090 11 378 378 378 378 1165 1627 1165 1164 1221 1179 908 098 12 361 361 361 361 1167 1591 1167 1179 1233 1206 947 1041 005 .ul 3 006 .tg NI 009 Table 2.2 006 .sp 1c 083 Values for the coefficients Aij and Cij in Scheraga's*S21 *Nparameterisation of the 024 Lennard-Jones potential. 003 .bp 006 .ls 3c 006 .tg NI 005 .ul 1 025 References for Chapter 2. 006 .tg NA 058 1. Weintraub H.J.R. *IPh.D. Thesis *N(1975) Case-Western 028 Reserve University 067 2. Hopfinger A.J. *I"Conformational Properties of Macromolecules" 033 *N(1973) Academic Press 053 3. Schrodinger E. *IAnn.Physik *D7_9_ *N(1926) 361 067 4. Pople J.A. and Segal G.A. *IJ.Chem.Phys. *D4_4_ *N(1966) 3289 059 5. See, for example, Pullman B. in *I"Quantum Mechanics of 058 Molecular Conformations"*N, ed Pullman B. (1976) 015 Wiley 065 6. Richards W.G. *I"Quantum Pharmacology" *N(1977) Butterworths 056 7. Murrell J.N. and Harget A.J. *I"Semi-empirical Self 054 Consistent Field Molecular Orbital Theory of 035 Molecules" *N(1972) Wiley 066 8. Claverie P. in *I"Intermolecular Interactions - From Diatomics 056 to Biopolymers"*N, ed Pullman B. (1978) Wiley 042 9. Richards W.G. *IPrivate Communication 062 *N10. Hoyland J.R. in *I"Molecular Orbital Studies in Chemical 055 Pharmacology"*N, ed Kiel L.B. 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Pullman A. and Pullman B. *IProc.Nucl.Acid.Res. and 038 Mol.Biol. *D9_ *N(1969) 327 064 24. Olsen W.K. and Flory P.J. *IBiopolymers *D1_1_ *N(1972) 25 002 *?