Materials Chemistry Research Dept. BL0111780 Research Paper

Frank H. Stillinger

Exponential Multiplicity of Inherent Structures

Frank H. Stillinger

Bell Laboratories, Lucent Technologies Inc.
Murray Hill, NJ 07974
and
Princeton Materials Institute, Princeton University Princeton, NJ 08544

January 29, 1998

Abstract: The mechanically stable spatial arrangements of interacting molecules (potential energy minima, ``inherent structures'') provide a discrete fiducial basis for understanding condensed phase properties. Simple plausibility arguments have been advanced previously suggesting that at fixed positive density the number of distinguishable inherent structures rises exponentially with system size. A more systematic analysis is presented here, using lower and upper bounds, that leads to the same conclusion. Further examination reveals that the characteristic exponential rise rate for inherent structure enumeration diverges as density approaches zero, when attractive interparticle forces are present.

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                          Frank H. Stillinger
                            January 29, 1998

                          Murray Hill, NJ  07974
                                   and
           Princeton Materials Institute, Princeton University
                           Princeton, NJ  08544


            The mechanically stable spatial arrangements of

       interacting molecules (potential energy minima, ``inherent

       structures'') provide a discrete fiducial basis for

       understanding condensed phase properties.  Simple

       plausibility arguments have been advanced previously

       suggesting that at fixed positive density the number of

       distinguishable inherent structures rises exponentially with

       system size.  A more systematic analysis is presented here,

       using lower and upper bounds, that leads to the same

       conclusion.  Further examination reveals that the

       characteristic exponential rise rate for inherent structure

       enumeration diverges as density approaches zero, when

       attractive interparticle forces are present.


                                  - 2 -


       I. Introduction  

            One of the intrinsic difficulties faced by the field of

       nonlinear optimization is that many problems of interest

       present large numbers of ``false'' solutions.  In the case

       of an objective function requiring minimization, the global

       absolute minimum may be hidden as a needle in a proverbial

       haystack of local nonabsolute minima, possibly requiring an

       exhaustive search and comparison procedure.  Indeed many

       families of problems are known for which the total number of

       minima rises at least exponentially as the number of

       variables increases.1



            Under some circumstances it may be valuable to identify

       and classify the entire collection of minima from the

       ``best'' to the ``worst,'' _i._e. from the absolute minimum to

       the highest-lying local minimum.  This is the case in

       condensed matter physics/materials science where the

       objective function in one important application is the

       potential energy of interaction F for the constituent

       particles, and its minima represent the mechanically stable

       arrangements of those particles in space (``inherent

       structures").2,3 If the particles involved number N, and are

       structureless, F would have to be minimized over the 3N-

       dimensional space of particle positions, rrrr1 . . . rrrrN.  If

       each particle additionally possessed @ internal degrees of

       freedom (describing orientation, vibrational amplitudes, or

       conformation), the relevant configurational space over which


                                  - 3 -


       F would have to be minimized would have dimension (3+@)N.


            Let Z(N,V) be the number of F minima when N particles

       are confined to volume V of given shape.  For a single

       component system (all particles identical) it is useful to

       write


                                  Z(N,V) = N!Z1(N,V) .               (1.1)



       This accounts for the fact that with hard walls present each

       minimum is but one of N! equivalent minima that differ only

       by permutation of identical particles.  [When periodic

       boundary conditions are imposed on V, the resulting free

       translation requires that the N!  factor in Eq. (1.1) be

       replaced by (N-1)!.]  Consequently Z1 only enumerates

       geometrically distinct minima.



            It has been argued2,3 that for realistic model

       potentials F that Z1 asymptotically rises exponentially with

       system size N (N/V > 0 held fixed, shape of V held fixed).

       More precisely, the claim has been that


9                                  N->oo8lim (N-1lnZ1) = A ,
(1.2)
9
                                        A > 0 .



       The exponential rise rate parameter A is expected to be

       substance-specific, and to depend on number density N/V.

       The tentative validity of relation (1.2) rests partly on the

       fact that some exactly solvable many-body models indeed
9


                                  - 4 -


       exhibit just that property.4,5 But it rests as well on a

       frankly crude and intuitive (but general) argument that

       macroscopic subvolumes of V could be geometrically reordered

       essentially independently of one another, and thus that Z1

       would have to be multiplicative over those subvolumes.6 The

       purpose of the present work is to supply a stronger general

       basis for the claim of exponential multiplicity of distinct

       inherent structures in material systems.



            The following Section II establishes on physical

       grounds a lower bound for Z1 that itself rises exponentially

       with N, so A in the right member of Eq. (1.2) must be

       greater than zero, if it exists.  Section III establishes

       that this right member is bounded above, using the strategy

       of _r_e_d_u_c_t_i_o _a_d _a_b_s_u_r_d_u_m.  Section IV
takes up the question

       of enumerating inherent structures in free space, and

       concludes that if attractive forces are present (as is true

       for ``real'' material systems), then A must diverge to

       infinity as N/V goes to zero.  Section V presents several

       concluding remarks, including some directed to polymeric

       substances and to mixtures.


                                  - 5 -


       II.  Discussion



            As in the preceding Introduction attention will focus

       for the moment on the single-component case.  Realistic

       interaction potentials F that describe such systems are

       continuous and at least once differentiable away from

       nuclear confluences; furthermore they are bounded below by

       -BN for some B > 0.7 In the large system limit of interest

       here, the absolute minimum of F will correspond to some

       periodic crystal structure whose details (symmetry, unit

       cell dimensions, etc.)  reflect molecular shape and

       flexibility, and the balance between intermolecular

       attractions and repulsions.  Several alternative, less

       stable, crystal structures may also exist for the pure

       substance of interest, however only the classical ground

       state (the absolute F minimum) need be considered for

       establishing a lower bound to lnZ1.



            Place the N molecules into one of the permutationally

       equivalent absolute-minimum configurations .  The

       resulting elastic solid may or may not entirely fill the

       finite available system volume V, depending on how the

       latter compares with the zero-pressure, zero-temperature

       volume of the N-molecule crystal.  In either event let V0 be

       the volume actually occupied by the crystal.


                                  - 6 -


            Divide V0 into identical compact subvolumes v0, of

       microscopic size, each containing on averge n0 molecules.

       The number of such subvolumes is


                                     V0/v0 = N/n0 .                  (2.1)



       The intention is to choose v0 sufficiently large (though

       still on the molecular scale) that a mechanically stable

       defect-containing rearrangement of molecules could be

       effected in each subvolume, without affecting the

       possibility of such rearrangment in any other subvolume.

       The type of crystal defect involved can vary according to

       the substance under consideration.  In the case of atomic

       substances a nearby vacancy-interstitial pair (Frenkel

       defect) is the natural choice, resulting from lengthwise

       displacement of a short line of particles.8 On the other

       hand, substances composed of large flexible molecules admit

       defects resulting from single molecular reorientation or

       internal motion.9



            Notice that we do not require the defects in separate

       subvolumes be noninteracting, but only that the interactions

       be sufficiently weak that the absence or presence of defects

       in all subvolumes be possibilities that are independent of

       one another.  Elastic strains surrounding defects will

       propagate through the crystal medium causing defect-defect

       interactions, but these strain fields die off algebraically

       with distance.10 Hence the independence assumption will



                                  - 7 -


       place a lower limit on v0 (and thus n0).



            Let \ be the number of distinguishable configurations

       that the defective state in v0 can adopt.  This might count

       the different relative positions of a vacancy-interstitial

       pair, or the different ``unnatural'' but mechanically stable

       reconfigurings of a flexible molecule.  In any event the

       number of undisturbed plus defective states considered for

       each subvolume is 1 + \.  On account of subvolume

       independence, we thus consider the following number of

       distinguishable, mechanically stable configurations

       (inherent structures) for the N-particle system:


                           (1+\)8V0/v09 _= exp{[n08-19ln(1+\)]N} .
(2.2)



       Presumably this represents only a small subset of all

       distinguishable inherent structures for the N particles in

       fixed finite volume V, so we can write


                               exp{[n08-19ln(1+\)]N} _< Z1 .
(2.3)



       If the limit indicated earlier in Eq. (1.2) indeed exists,

       then this last expression implies


                                  0 < n08-19ln(1+\) _< A .
(2.4)

                                  - 8 -


       III.  _U_p_p_e_r__B_o_u_n_d



            The next task is to examine implications of possible

       violation of the limiting behavior in Eq. (1.2) due to

       greater-than-linear rise of lnZ1 with N.  Suppose

       tentatively that the following large-N behavior (with

       positive N/V fixed) applies:


                                     lnZ1 9~8 f(N) ,                   (3.1)



       where


9                                  N->oo8lim [N/f(N)] = 0 .
(3.2)
9


       This could arise, for example, if f(N) were proportional to

       Nq, q > 1.  Such behavior has significant consequences for

       the mean size of basins belonging to the system's inherent

       structures.



            The configuration space content for a single

       molecule/particle can be written as VC.  The first factor is

       attributable to center of mass translation, while the second

       factor is just the integral (between bounded limits) of the

       @ internal degrees of freedom.  In the simple case of

       structureless particles (@=0), C is set to unity.  The

       content of the multidimensional configuration space

       describing all N molecules/particles simultaneously is

       (VC)N.  The mean basin content emerges upon dividing this

       content by the number of basins:
9

                                  - 9 -

99
N!Z1(N)7(VC)N9_______ .                     (3.3)
9


            In order to interpret the last expression physically,

       it is useful to re-express it in terms of a mean linear

       displacement scrL for each molecule/particle.  Consequently

       (3.3) can alternatively be written in the form (CscrL3)N.

       The asymptotic large-N behavior tentatively postulated for

       Z1 then leads to the following result for scrL:


                            scrL3(N) 9~8 (V/N)exp[1-f(N)/N] .          (3.4)



       The postulated property (3.2) for f(N) forces scrL to vanish

       in the large system limit (N -> +oo, positive N/V fixed).

       This is physically unacceptable because it implies that

       arbitrarily small displacements in virtually any direction

       suffice to switch the system from one inherent structure to

       another. In particular this would render impossible phonon motions
of finite amplitude in the crystalline state (no restoring
forces), as well as kinetic arrest in nonergodic trapped
glassy states (a common occurrence for amorphous substances). Conventional experience however indicates that

       scrL should remain positive and of the order of molecular

       dimensions in the large system limit.  This can only happen

       if f(N) is linear in N, and in accord with the earlier

       Eq. (1.2), specifically:


                                      f(N) 9~8 AN .                    (3.5)


                                  - 10 -


       IV.  _D_e_n_s_i_t_y__D_e_p_e_n_d_e_n_c_e



            The considerations presented in the two preceding

       Sections II and III force the conclusion that Z1 indeed

       rises exponentially with N at fixed positive density, _i._e.

       that A in Eq. (1.2) is well defined.  However this leaves

       open the issues of how A depends on the substance under

       consideration, and for any given substance how this

       parameter varies with density.



            A particularly simple situation arises if the potential

       energy function F is homogeneous of degree -n, n > 3.  This

       obtains specifically when F is composed of purely repulsive

       inverse-power pair potentials:


                             F(rrrr1 . . . rrrrN) =9 i +oo.  The very open and multiply branched

       structures produced by diffusion-limited aggregation

       processes18 suggest that similar forms might be expected for

       free-space inherent structures.  This in turn invalidates

       the basis on which the exponential rise rate of Z1 with N,

       Eq. (1.2), has been established for fixed positive density.



            An elementary, approximate, enumeration scheme for

       inherent structures in free space implies that Z1 rises more

       rapidly than as a simple exponential.  Suppose in fact that

       the ``typical'' free-space inherent structure is indeed very

       open.  Imagine constructing such arrangements particle by

       particle from an initial seed.  At any intermediate stage,

       the number of available sites at which the next particle

       could be attached to the incomplete cluster would be roughly

       proportional to N9'8, the number already in place.  Therefore

       we have (K > 0):


                                 Z1(N9'8+1) ~= KN9'8Z1(N9'8) .
(4.2)



       Taking logarithms, and treating the large variable N9'8 as

       continuous, we have


                                  - 13 -

                               dlnZ1(N9'8)/dN9'8 ~= ln(KN9'8) .
(4.3)



       This can be integrated to yield the following:


                             lnZ1(N) ~= NlnN + (K-1)N + C ,           (4.4)



       where C is an integration constant.  Clearly this result

       contradicts the positive density presumption embodied in

       Eq. (1.2), and suggests a faster-than-exponential rise rate

       with increasing N.



            A more insightful enumeration scheme than the crude one

       just presented might reveal that as particles are added, the

       previously emplaced ``substrate'' might not simply serve as

       a nearly rigid host, but be capable of new and distinct

       stable arrangements that could not exist without the

       additional particle.  If such possibilities are present and

       sufficiently numerous, the estimate (4.4) above might

       actually be a significant underestimate.  That could

       conceivably lead to the form (3.1) shown earlier with f(N)

       proportional to Nq, q > 1.  But without having to settle

       these technical details, we can safely conclude that for any

       three-dimensional model substance possessing attractive

       interparticle interactions, parameter A must diverge to

       infinity as the density goes to zero.


                                  - 14 -


       V.  _D_i_s_c_u_s_s_i_o_n



            The lower bound for A provided by Eq. (2.4) may prove

       to be very weak in many applications.  In order to satisfy

       the defect-independence assumption on which that result is

       based, n0 would probably have to be of order 102; assigning

       the value 6 to \ then might be reasonable.19 Consequently

       Eq. (2.4) would state


                           (ln7)/100 = 0.0194591 . . .  _< A .        (5.1)



       By contrast, Wallace20 estimates that


                                        A ~= 0.8                      (5.2)



       for a wide range of monatomic substances.  Flexible organic

       molecules such as the fragile glass former ortho-terphenyl

       (OTP) appear to exhibit substantially larger A values; a

       simple calculation based on its measured heat capacity and

       heat of fusion suggests that21


                                    A(OTP) ~= 13.14 .                 (5.3)



       Linear polymeric substances may exhibit A's that increase

       roughly linearly with degree of polymerization (numbers of

       monomer units), at fixed overall mass density, owing to

       backbone flexibility degrees of freedom.  These

       discrepancies with Eq. (5.1) warrant searching in the future

       for more powerful bounds for A.



                                  - 15 -


            The focus of the foregoing arguments has been the total

       number of inherent structures, regardless of how they may

       differ in detail.  But it is also important to classify

       inherent structures according to one or more intensive

       ``order'' parameters, and if possible to obtain their

       distribution with respect to these parameters.  A

       particularly important case involves U, the potential energy

       per particle of the inherent structures, because this leads

       to an especially simple expression for the free energy of

       the many-particle system.3,6 Given the validity of

       Eq. (1.2), and assuming the continuity of the asymptotic

       distribution with respect to U, it is inevitable that this

       distribution of distinguishable inherent structures have the

       form3,6


                             exp[NY(U)]      ,      Y _> 0 .
(5.4)



            Although the developments in the preceding

       Sections II-IV have been restricted to single component

       systems, mixtures also deserve examination.  In the general

       case involving components 1 . . . @, Eq. (1.1) generalizes

       to

                     Z(N1 . . . N@, V) = (PNM!)Z1(N1 . . . N@, V) ,
(5.5)


       with Z1 expected asymptotically to rise exponentially with

       the total number of particles (all densities NM/V held

       fixed).  But because the components are distinguishable, the

       exponential rise rate quantity A should be larger than its


                                  - 16 -


       single-component relatives on account of mixing entropy

       effects.


            A concrete example serves to illustrate the last point.

       Let A(N2) be the exponential rise rate quantity for pure

       molecular nitrogen.  Carbon monoxide, CO, has a small

       molecular dipole moment,22 and has nearly the same molecular

       size as the dipole-moment-free nitrogen molecule.23

       Consequently CO should be able freely and stably to

       substitute for N2 in any inherent structure for the latter.

       Taking due account of the two distinguishable orientations

       available for each substituting CO, one estimates A for the

       N2 - CO mixture to be


                        A = A(N2) + xln2 - xlnx - (1-x)ln(1-x) ,     (5.6)



       where x is the mole fraction of CO in the mixture.


                              References

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         4.  F. H. Stillinger, J. Chem. Phys. 88, 380 (1988).

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