NEW FITTING M ETH O D (»LP FITTING METHOD*)

The LP fitting m ethod is designed to fit to ab-initio surfaces w ith the a d d e d co n strain t of req u irin g observed ex p erim en tal stru c tu re s to reproduced. Some ideas are proposed in order to overcom e the problem s described in Section 5.5.1.

First, regarding the problem of finding the global m inim um , if the problem can be linearized, the global m inim um can be found w ith in a finite n u m b er of iterations. The second p o in t is th at se p aratin g the

criterion of crystal stability from the evaluation of properties can m ake the fitting problem m uch easier. It is also desirable th at the experim ental d ata (such as the structural stability conditions) should be separated from the ab-initio potential energy data as com ponents of the cost function. On the o th er h an d , w hen com m on potentials th a t can rep ro d u ce several different stru ctu res at the sam e tim e are desired, the in tro d u ctio n of in d ep e n d en t sets of stru ctu ral stability conditions is m ore reasonable th an the use of a unique form ula of S. The th ird p o in t is th at it is very helpful to know w hether a solution of the problem is feasible or not, and also w hich condition obstructs the solution. In particular, it is n o t clear how w ell the B2O3 system w ith covalent bonding can be described w ith

the existing potential functions.

These ideas lead to new p otential fitting m eth o d based on the L inear P ro gram m ing (LP) m ethod. The LP m eth o d is a w ell-know n technique in the field of economics and m athem atics (see D untzig 1963).

Several special considerations are given in o rd er to a d a p t the potential fitting problem to the general LP problem as follows:

i. All the conditions are separated into tw o categories: one is several sets of inequality equations; the o th er co m p o n en t is a cost function w hich should be m inim ized. The fitting problem is replaced w ith the problem w hich finds the optim um solution th at m inim izes the cost function w ith in the solution space th at satisfies all the inequality equations.

ii. The conditions of structural stability are defined in the form of inequality equations. Here, the term 'structural stability' m eans th at the relaxed structure does not distort m uch from the experim ental structure. The lattice energy in the experim ental stru ctu re is th o u g h t to be the m inim um point in the configurational space (3N-dimension) of energy.

The lattice energy E is defined:

E = E (xi,x2, • • •, xn), (5.6)

w here the xi show the position vectors of the i-th atom.

3 E /a x i = 0 ; a 2 E /3 x i2 > 0 (5.7) at the experim ental structure ( x i= x ie ,. . , xn=xne)*

then '^Axi :

E (xie, X2e, • •, xie+Axi,. . , Xne) > E (xie,x2e, • •, x ie ,. . , Xne) (5.8) an d

E (xie, X2e/ • - , xie -Axi,. . , Xne) > E (x%e,x2e, • •, xie, • • / Xne) (5.9) m u st be satisfied. W hen lattice param eter a,b,c,a,P and y (6 variables) and

n atomic positions (n x 3 variables) are taken into account, total (6n + 12)

inequality conditions are generated.

iii. The w eighted sum of the residuals betw een the ab-initio data on the p o ten tial energy surface and the estim ated values u sin g the u n k n o w n v ariab les of p o ten tial p aram eters is d efin ed as the cost function. Therefore, the LP m ethod tries to find the solution w hich realizes the global m inim um residual w ith in the so lu tio n space th at satisfies the stru c tu ra l stability conditions. H ow ever, th ere are tw o lim itations for applying the LP m ethod. O ne is th at it is n o t easy to im p lem en t the evaluation of various physical p ro p erties as the cost functions, because w ith in the LP schem e the com plex form of such properties m ust be linearized using the potential param eters. The other is th at the cost function m ust be the linear w eighted sum of potential p aram eters instead of the square w eighted sum of them . But there is little difference betw een two ways of sum m ation.

The m ost im portant point of the LP m ethod is th at the structural stability conditions are not included in the cost function, b u t in the inequality equations. Therefore, the m erit of this m ethod is that, even if

the p otential functions are poor in describing the structure, it finds a so lu tio n w hich m aintains the stru ctu re in eq u ilib riu m , o th erw ise it returns the message saying that there is no feasible solution.

iv. Once the problem is described w ith in the fram e of LP, the so lu tio n is quickly solved even by a p e rso n al com puter. The m ost significant problem w ith this m ethod is th at the fitting problem m u st be linearized regarding the potential param eters. Inevitably some p arts (e.g. p p aram eter in Buckingham form in (5.2), or pij and ro p aram eters in M orse form in (5.3)) cannot be linearized sim ply and m u st rem ain fixed as constants d u rin g one solution cycle. H ow ever, each solution cycle is very quick, allowing a thorough search of a variety of com binations of p, Dij or pij to be easily perform ed, in order to find the global m inim um .

The algorithm used by the LP fitting m ethod is show n as below:

i. L inearization of each of the term s of w hich the total lattice energy E is comprised:

E = Ec + E2 + E3 + E4 (5.10)

w here Ec, E2, Eg and E4 are the contribution of Coulombic energy, pair-

potentials, three-body term s and four-body terms.

For Coulom bic term s, they are calculated only from the crystal stru c tu re w h en the charge values are fixed, and th ey are d ealt as constants in inequality equations.

In case of Buckingham form,

E2 = Si>j (Aexp(-rij/p) - C /rij6}

= A • Ei>j exp(-rij/p)} - C • {Ei>j l/rij^} (5.11) The values in the parentheses { } are calculated only from the crystal stru ctu res, and are in d ep en d en t of the u n k n o w n variables A an d C, w hen p is fixed.

In case of Morse form,

E2 = Dij • Si>j [exp{2Pij(rij-ro)} - 2exp{Pij(rij-ro)}] (5.12) Once m ore the value in the parenthesis [ ] is calculated only from the crystal structures, and is in d ep en d en t of the un k n o w n p aram eters Dij w hen pij and ro are fixed.

For the sim ple harm onic-type three-body term s,

E3 = Kb • Ei>j > k(l/2 (0ijk - 00)2} (5.13) The value in the parenthesis { } is again calculated only from the crystal structures, and is independent of the u nknow n param eters Kb w hen Go

is fixed. Four-body term s are dealt in the sam e w ay as the three-body term s.

ii. To set up the inequality conditions for the structural stability: E ( x i e ,. . , xie±Axi,.. ,xne) > E (xie, • •, xie, • • / xne) (5.14)

The coefficients of the unknow n variables (A, C, Dij and Kb etc.) are calculated for each structural configuration, and (6n +1 2) sets of inequality

equations are generated. For example,

C14 • Db_o + C1,2 • Ao-O - Cl,3- Co-O + Ci,4 • Ko-B-O > 0

C2,l • Db-O + C2,2 • Ao-O - C2,3 • Co-O + ^2,4 * Ko-B-O > 0

C6n+1 2,l ' Db-O + C6n+1 2 ,2 * Ao-O ‘ C6n+12,3 • Co-O + Cén+UA ' Ko-B-O > 0 In the case of the B2O3 system, the Morse form for the B-O interaction, the Buckingham form for the 0 - 0 interactions, and the three-body term for the O-B-O interactions are used, and there are four variables (D g- O /A O -O /C O -O and KQ-B-O)* A m ong the inequality equations, tw elve come from the variations of the cell param eters (a±Aa, b±Ab, c±Ac, a±A a, p±Ap, 7±Ay), an d 6n come from tha variations of internal coordinates

iii. To define the cost function S:

The deviation betw een the ab-initio d ata on the p o ten tial energy surface and the corresponding value estim ated from the linearized sum of the param eters are sum m ed up as the cost function S.

S = I iW i- lE iab-E ies I (5.15)

w h e re w , is the w eighting factor, and Ei®b and Ei^s are the energies derived from the ab-initio calculations and the estim ated energies as the linear sum of the potential param eters. U sually all the w eighting factors are set to 1, and do not need to be changed.

iv. To add the extra inequality conditions if necessary:

For exam ple, if the total lattice energy is restricted w ith in som e specific range (for exam ple, Em in < E and E < Emax are given), tw o inequality equations are ad d ed in the sam e m anner as in (ii). It is also very easy to specify the difference of energies betw een several different stru ctu res (for exam ple, w hen the energy differences AE12 and AE23 betw een three polym orphic structures are given, E% + AE1 2 < E2 and E2

+ AE2 3 < Eg are a d d e d ).

V . To apply the general LP algorithm:

The coefficients calculated from i. to iv. generate the general m atrix elem ents for LP and the variables are solved so th at they m inim ize the cost function S at the finite calculation steps.

vi. Iterate from (i) to (v), changing the non-linear p arts (e.g. p, p, or ro) to yield the solution w hich realize the global m inim um . As m any co m b in atio n s as possible of th e u n k n o w n p a ra m e te rs are a p p lie d systematically.

We m ay com pare this LP fitting m eth o d w ith the o th er general algorithm s as follows:

The strengths of the m ethod are:

i. It is especially suitable for the ill-conditioned problem , w here the crystal stru c tu re is a p t to m ove to w ard a catastro p h ic change (for exam ple, in the case of lay ered or p lan a r stru ctu re). Because the structural stability conditions are absolutely satisfied du rin g the solution, it can alw ays prevent the distortion of its structure. The m ethod is also suitable for the sim ultaneous fitting am ong several structures, because all the stru ctu ral stability conditions are satisfied in d ep en d en tly and sim u ltan eo u sly .

ii. W hen the linearised coefficients are output, the potential energy surface w hich depends on the variables (A, C, Dij and Kb) can be easily

analyzed, because it is sim ply the linear sum of each term . In particular, w hen a satisfactory potential cannot be obtained, it is straightforw ard to find w hich stability condition obstructs the solution.

iii. The global m inim um can be ob tain ed w ith a v ery m o d est com puter resources. There is no problem about setting initial conditions or the w eighting factors, and no empirical adjustm ents necessary.

The w eak points of this m ethod are;

i. The m ethod cannot be applied generally. The requirem ent th at all the conditions m u st be linearized is v ery restrictive. T herefore, features including fitting to crystal properties or use of the shell m odel cannot be included at the m om ent. In such cases it is possible to refine the LP-fitted param eters by u sing m ore general fitting program s. It is in te re s tin g to n o te th a t th is LP m eth o d is b a se d o n lin e a riz e d optim ization w ith constraints, w hile the other general m ethod is based on n o n -lin ear o p tim izatio n w ith o u t constraint. In fu tu re, w h e n LP m ethod is iteratively solved on one hand and the other general m ethod

a d d s the constraint conditions on the other h an d , b o th m ethods w ill approach one another.