Optimization of parameters for an effective hamiltonian

Optimization of parameters for an effective hamiltonian

Volume 60, number 3 CHEMICAL. FHYSICS LE.l?I-JZRS 15 January1979 OPTIMIZATION OF PARAMET ERS F!3R AN EFFECTIVE HAMIL’I’ONIAN William C_ HERNDON and...

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Volume 60, number 3

CHEMICAL. FHYSICS LE.l?I-JZRS

15 January1979

OPTIMIZATION OF PARAMET ERS F!3R AN EFFECTIVE HAMIL’I’ONIAN William C_ HERNDON and M. Lawrence EILZEY Jr. Deputrnent of Ckmht.?y, Universi~ of Texas at El Paso, El Paso, Texts 79968, USA

Received 19 September 1978

A generaIIy applicable, least-squares method that can be used to optimize the parameters of an effective hamiltonian matrix in any basis set Is dexxibed. The technique is applied to the correIation of photoelectron spectra of saturated hydrocarbons.

1. introduction The parameterized hamiltonian matrix is one of the most useful conceptual modeis for correlating and discussing a wide variety of chemical topics. One wellestabhshed application is the interpretation of the photoelectron spectra of organic compounds in terms of Hilckel pielectron theory*_ A second interesting use ti the construction of effective Fock matrices, assuming that mean zh~es of calcuiated Fock matrix elements can be ccnsidered as transferable parameters [Z--5] _ The methods used for parameterization are not uniforrnIy estabfished. For exampIe, consider the highly pararneterized localized orbital quantum mechanical calculations used to correlate ionization potentials (IP’s) obtained from photoelectron spectra (pES) of aikanes f6--16]+*_ The parameters that need to be determined are the CC and CH bond diagonal matrix elements, and the off-diagonal interaction terms of the effective molecular hamiltonian matrix. One general approach is to mdy symmetrical model compounds that contain at least one symmetry-adapted moiecular orbit& for which the eigenvaIue (-IP) is a linear function of some of the parameters_ Murreil and Schmidr [IO] combined the results of ab-initio calcul See, for example, the work of Heilbronner and his collabo** rators. A leading reference is ref. i I]_ For a review of work previous to 1970,seeref. [6].

510

with an analysis of this type to correlate successfulzy the PI3 of saturated hydrocarbons. -4 more empirical method, used by Bra&ford and Ford [7], employs colnputer-search procedures where each separate parameter is varied till a final optimum set is generated. Also, Heilbronner et al_ [ 1S] , were highly successful in correlating inner vatence molecular orbital energies corresponding to carbon 2s ionization potentiaIs with a simple two-parameter topoIogicaI equivalent orbital method. In this latter application the best parameters were found by a least-squares technique. We have previously described a method based on matrix algebra that requires a complete knowledge of the experimental PES data and symmetry-adapted eigenvectors. The process uses a reverse similarity trausformation to generate the complete hamiltonian matrix of diagonal and interaction terms for each molecule. The mean values of matrix elements for similar environments are then used as parameters for subsequent bond orbital caIculations. The method requires a large number of iterations before parameters are obtained that do not vary in successive rriaIs (5 significant figures), but gives au excellent correlation of ca.IcuIated and experimental PFS data (correlation coefficient 0.9995) [16]. The purpose of this paper is ro descriie a more general procedure that can be used to optimize an effestive ha.rniItonianmatrix in any basis set. This procedure uses &t-order perturbation theory to write simultaneous linear equations iu the unknown matrix elel&ions

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CHEMICAL PHYSICSLETFERS

ments, and a standard regression analysis gives the parameters.

2. Linear regression procedure

ln th-is sectionwe discuss the optimization procedure employed in this work. The effective hamiltonian operator is assumed to be expressible as a linear combination of m operators

(1) The operators Mi are completely defined over an appropriate basis of the state space, and the coefficients pi are parameters to be evaluated from experiment. In terms of matrix representations over an orthonormal basis, eq. (1) becomes (klH(Z)

= zgI pi~klMilZ~ _

of eigenvalues to energies and any other physical or theoretical conditions. The number of observations m must equal or exceed the number of parameters. In order to develop the iterative procedure we perform a first-order expansion analogous to Newton’s method for roots. Let the parameters of the ti iteration be

pw rpF-1) z I

then the residti R<‘) = R<‘-l) I I

-E[IJ)=

0

6Ej 6 ~=QEjlHIEj) I i 6(Ejl

-=-iE)+ =(Ei16pi i --HIEj)f api

For a particular set of parameter vahres, the calculated energies Eg will diffe- from a set of n observed energies 5’ by -

‘EiJHJEj)_

(E’lH$lEj) I

IE”‘Ejl&lEjb

r

- (91

The quantityin parentheses is the derivative of CE’ Iaj_’ = 1, and hence is zero_ Therefore = (EF-“IM&Ej(t--l+ .


Substitution of eq. (10) into eq. (8), using eq. (7), gives

where IEi) is the normalized eigenvector corresponding to Ei- A special case of eq. (4) is a well-known relationship in Hiickel MO theory 1171 that gives the orbital energies in terms of the Coulomb integrals (1~and resonance integrals 8:

-Ei=E’

)

now

6Ej/6pilpcf_l)

PiWjlMilEj) 3

_ ai

1Mi !Ei(t-l)

(3)

m Ej=(EiIHIE’~~z~l

Ri=E;

f dRj = R:f-l’

(2)

are to be identified with the energy levels of the system under consideration. Note that

(7)

are

= E? _ zp’f-l)q-” 1 i i

Eigenvalues Ej of H obtained from the secular equation det([H]

+ &Is )

(6)

R:”

= E; -

c

(py-”

+ dpi)(&“-l)lNl,lEj(t-l))

i =E;

-

&,ta(E~-1)IP.4ilEi(t-‘)). i

(11)

For n = m, the residuals at each step can be set equal to zero and eq_ (11) solved for the unique parameters, asmming the rank of the coefficient matrix to be m_ For n > m, the sum of squares of the residuals is minimized by linear regression for the py) [18,19].

3. Parameterization using PES

Aikoptinnnn set of parameter values rninhn&es the residuals, ZQ, consistent with rhe correct correlation

To exemplify the procedure outlined in the previous 511

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section, PES data for the .=turated hydrocarbons, methane, ethane, propane, isobutane, and neopentane, are used to parameterize *he localized bond orbital effective hamiltonian. We utilize the eXperimental PES results that are obtained from He 58.4 and 30.4 nm rad%ttion [ 10,20] _ The parameters to be obtained are defined as shown below for propane. In nnmber and in type, they are the same as those considered in previous work, except for long-range parameters 6 I and Gq (neglected by Murrell and Schmidt [lo] and Brailsford and Ford [7]), and g3 and 6, (not differentiated by Herndon et al_ [ 161). Long-range parameters are XIV to lift accidental degeneracies in isobutane and neopentane.

Js-

, -at

(LO

a2

CALCUlAlED [email protected]

3

9

#11=<11H12)=(21H13)=(5IHI6), & = ; =<11H15>=<3lalS>, = t2lEl5>, = <2]n]7), = (1 lUl7>;

6, =(l IHl8>, 62 = <2lHl l), 63 = (1 fHl9>, 6, = <2lHl9)_

(12)

Fig_ 1 is a graph of the eigenvalues of Sb versus the experimen*M vertical P’s_ The correlative ability of this simple overlap model is remarkable and was un512

URBITRARY

1

WITS1

potent&Is of methyhethanes

(overlap basis).

Table 1 parameters for bond orbital CalcuMions

Initial sets of eigenvectors were obtained by solving the CC and CH bond Slater overlap matrix Sb for each molecule. The bonds were expressed as linear combinations of hybrid carbon sp3 and hydrogen 1s orbitals. The CC and CH bond lengths were taken as 1.54 and 1.09 A respectively. The small difference between the CC and CH diagonal matrix terms was assumed to be proportional to A = 1 - Sa/S,. Diagonalization of Sb yields symmetry adapted eigenvectors T and eigenvalue3 A in arbitrary units: h= TSbT.

Ionization

as

expected- However systematic deviations from a linear relationship are evident, even though the correlation coefficient of the overlap model with experiment is 0976 (standard deviation of calculated II% = +1 .O ev). The eigenvectors in the bond overlap basis are then used as starting vectors to obtain the matrix elements of the bond orbital effective molecular haruiltouian. Computer time (IBM 360-50) required for the iterative operation is 15 s (7 iterations)_ Identical fmal results

al (CH bond diagonal term), o2 (CC bond diagonal term);

ri r2 y3 74

1.

(lb

a4

M&ix

element

DWXiptiOIX

Linear

regresion -16.017 -15.953

diagorla term IiiagonaI

term

geminal intern&on gem&l interaction geminal interaction

-2.312 -1.978 -1.904

vicinal vivichai vi&al

-0.650 i-o.975 -0.466 i-o.725

gauche interaction trans interaction gauche interaction trans interaction

I$-interaction 1,4-semction 1,4-iuteraction 1.4-htem~on

(w-type) (ck) (skew)

-0.449 -0.209 +0.116 +0.005

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Table 2 Ionization potentials of saturated hydrocarbons Compound

Symmetry species

IP’s (eV) calculated

PES

methane

1% 2=,

13.7: 22.95

14:o 22.9

ethane

leg 3a1g leu 2a2, 2a1g

12.08 13.01 15.33 20.32 23.91

12.1 13.0 15.4 20.4 23.9

2b2

11.51 11.96 12-17 13.49 13.89 15.28 16.11 19.33 21.91 24.41

11.5 12.1 12.5 13.5 14.1 15.3 16.0 19.4 22.1 24.5

2e

11.50 11.67 12.82 13.56 14.95 16.14 18.53 21.93

11.2 11.8 12.8 13.4 14.9 16.0 18.4 21.9

3%

24.80

24.8

4t2

1%

11.45 12.82

11.3 12.7

:;

15.44 14.17

15.4 14.i

4% 2% 3%

17.55 21.96

17.6 21.9

25.14

25.1

propane

6% 4bI 1% 3%

5% lb2

4a1 2% 3a1 isobutane

6% 52 la2 4e 3e 5%

4%

neopentane

are obtained using startingeigenvectorsbased solely on the topological and graph-theoreticalanalyses of these systems given previously [ 161. Identical results are also obtained if we initiate the iterativeprocedure with the bond-orbital parametersof Bra&ford and Ford 171. These matrix elements are given in table 1. The parameterscan be used to calculate ll?s in good agreementwith experimentalvalues as listed in table 2, and plotted in fig. 2. The standarddeviation of a calculated IP is kO.15 eV. The theory-experiment line has unit slope, zero intercept, correlation coefficient

0 AtEWINE EEIRANE Pmpmf 0 ISOBUTANE l NEOPENTANE

a 15 0

13.0

130

150

17 0

19 0

CALCLlL4lELl IP’S Fig.

ZLO

a0

zsc

IeV)

2.Ionization potentials of methylmethanes @araineterized

calculations).

0.9995, and a comparison with fig. 1 shows that systematic deviationshave been eliminatedby the parameterizationprocedure.

4. Discussion Our intention in this article is to illustratethe procedure for parameterizationof an effective hamiltonian. An excellent correlation of the data is obtained, but many previous calculationshave also given very good results. The choice of experimentaldata, assignments of the irreduciblerepresentationsof the molecular orbitals, and the signs and relativesizes of the parametershave all been extensively discussedin previous publications [6- 161. The advantages of this parameterizationmethod are speed of application, generality, and the fact that all desired experimental or theoretical data are used simultaneously. One additional major advantageis that the requirementof a complete startingset of eigenvalues of PES spectra is eliminated_It is possible to utilize only the most precise data to establishparameters. In the example given, this factor was not important, since the spectra are relativelywell resolved for this set of hydrocarbons. However, for many other hydrocarbons, the photoelectron spectralbands are

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broad and diffuse. Only the extreme highest and lowest values may be unambiguously determined. With the present technique, these values can be included in the parameter&ration data set in a simple and straightforward manner.

Acknowledgement

This work was supported by the Robert A. Welch Foundation of Houston, Texas. Generous amounts of computer time were made availableby the University of Texas at EI Paso. Part of the work was carried out while one of the authors (W.C.H.) was a Program Direcror at the National Science Foundation, and reIeasedtime for researchwas made avaiIableby the Foundation_HelpN criticism by Dr. CT. Michejda is aIs0 greatryappreciated_

References [ 11 E. Heilbronner and J9. Maier, in: Electron spectroscopy: theory, techniques, and applications, eds. CR BnmdIe ud AD. Baker (Acxktic Press. New York, 1977) pp_ 205-292_ [Z) Ph. Degard, G. Leroy and D. Peeters, Theoret. Chim. AC-~ 30 (1973) 243.

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[3] G. Leroy and D. f&&e-

I5 January1979 Theoret, Chim_ Aeta 36 (1974)

[4] z-Depks, G_ Leroy and D. Peeters, Theoret Chim. Acta 36 (1974) 109. [S] B. OXeasy, B.J. Duke and JE Eikrs, Advan. Quantum (s6em. 9 (1974) 1. 163 WC Hemdon. Rogr. Phys. Org. Chem. 9 (1972) 99. [7] D.F. Bra&ford and B. Ford, MoL phys. 18 (1970) 621. [8] WC Hemdon. Chem. whys. Letters 10 (1971) 460. [S] A.D. Baker, D. Bettezidge+ N.R Kempand RE. Kirby, J. Mot Stmc+ 8 (1971) 75, [lo] J.N. MureIl and W. Schmidt, J. Chem. Sot. Faraday Trzz. II (1972) 1709. [ll] W. Schmidt and B.T. WZcins, Angew. Chem. tintern. Ed. ll(1972) 221. [12] S. Bhattacharyya and k Bhaumik, Intern. J. Quantum Chem. 6 (1972) 935_ [13] W. Schmidt, Tetrahedron 29 (1973) 2129. [ 141 P-N_ Dyachkov and AA. Levin, Theoret chia Acta 33 (1974) 323; 36 (1974) 18L Il.5 ] G. Biexi, J.D. Din, E. Heilbronner and k Scbmelzer, Helv. Chim. Acta 60 (1977) 2234. [16] W.C. Hemdon, M.L.. ElIzey Jr. and K. Raginweer, J. Am. Chem- sac. loo (1978) 2645. [ 171 B.H. Chiigwk and Ck Co&on. Rot Roy. Sac. A201 (1950) 196. 1181 W.C Hamilton, Statistics io physical science (The Ronald Pres, New York, 1964). [ 191 CL.. Lawson and RJ- Hanson, SoXving least squares pro& Iems (Pnmtic&kIl, Engiewood Cliffs, 1974). 120J kw. Potts and D.G. Smxts, J. Chem. Sot. Faraday Truu IX (1974) 875.