Volume 60, number 3
CHEMICAL. FHYSICS LE.l?IJZRS
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, leastsquares 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 [Z5] _ 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 f616]+*_ The parameters that need to be determined are the CC and CH bond diagonal matrix elements, and the offdiagonal interaction terms of the effective molecular hamiltonian matrix. One general approach is to mdy symmetrical model compounds that contain at least one symmetryadapted moiecular orbit& for which the eigenvaIue (IP) is a linear function of some of the parameters_ Murreil and Schmidr [IO] combined the results of abinitio 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 colnputersearch 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 twoparameter topoIogicaI equivalent orbital method. In this latter application the best parameters were found by a leastsquares technique. We have previously described a method based on matrix algebra that requires a complete knowledge of the experimental PES data and symmetryadapted 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 &torder perturbation theory to write simultaneous linear equations iu the unknown matrix elel&ions
Volume 60. number3
CHEMICAL PHYSICSLETFERS
ments, and a standard regression analysis gives the parameters.
2. Linear regression procedure
ln this 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 firstorder expansion analogous to Newton’s method for roots. Let the parameters of the ti iteration be
pw rpF1) 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(tl+ .
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 wellknown 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(tl)
(3)
m Ej=(EiIHIE’~~z~l
Ri=E;
f dRj = R:fl’
(2)
are to be identified with the energy levels of the system under consideration. Note that
(7)
are
= E? _ zp’fl)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(tl))
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
Volume 60, number 3
tlxiENIcALPHysIcs
15 Januvy
LJzrTERs
1979
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 longrange 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). Longrange 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 36050) 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 io.975 0.466 io.725
gauche interaction trans interaction gauche interaction trans interaction
I$interaction 1,4semction 1,4iuteraction 1.4htem~on
(wtype) (ck) (skew)
0.449 0.209 +0.116 +0.005
Vohhne 60, number 3
CHEMICAL
PHYSICS
JxrrERs
1.5Jamzary 1979
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 1217 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 graphtheoreticalanalyses of these systems given previously [ 161. Identical results are also obtained if we initiate the iterativeprocedure with the bondorbital 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 theoryexperiment 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
513
Volume 60, number 3
fzHEMIcAL
PHYSICSLETERS
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_ 205292_ [Z) Ph. Degard, G. Leroy and D. Peeters, Theoret. Chim. AC~ 30 (1973) 243.
514
[3] G. Leroy and D. f&&e
I5 January1979 Theoret, Chim_ Aeta 36 (1974)
[4] zDepks, 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 PN_ 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.