Configuration Interaction in Spectroscopy
that we deduced the same temperature for the glass point for dielectric measurements' of water absorbed in collagen. Usually the transition to a vitreous state is more abrupt. In the case of bulk water a glass transitionlg has been observed near -138 "C. The heat capacity at -138 "C, if water is a metastable liquid, is approximately 12 cal deg-l mol-' and at -143 "C, when it is in the vitreous state, its heat capacity is 5 cal deg-' mol-'. We note that the values of the heat capacity for bulk water in the vitreous and the crystalline state are approximately equal. In the liquid state, however, large configurational contributions exist that appear within the unusually sharp range of 5 deg on raising the temperature. The slow decrease of the partial molar heat capacity of water observed in Figure 2 could be interpreted as an extremely spread-out glass transition range from +30 to approximately -100 "C. That the glass transition is less abrupt may be a consequence of the one-dimensionality of the water chains in collagen, instead of the three-dimensionality of bulk water. It is a general feature of phase transitions that the degree of cooperativity is smaller in one- than in three-dimensional systems. Compare, for example, the gradual helix-coil transitions of polypeptidesz0to the sharp melting point of three-dimensional crystals. It is easy to understand why the glass point of water absorbed in collagen is higher than in bulk. According to NMR and dielectric data the rotational relaxation times of water in collagen are considerably longer than for bulk water. Apparently, the fixed triple-stranded helices offer great resistance to the mobility of water molecules. It is then not surprising that water molecules become immo-
The Journal of Physical Chemistry, Vol. 82, No. 14, 1978
1663
bilized at a relatively high temperature. Acknowledgment. We thank the National Science Foundation for supporting this research through Grant No. BMS-75-07641.
References and Notes A. Miller and D. A. D. Parry, J . Mol. Biol., 75, 441 (1973). M. A. Rougvie and R. S. Bear, J . Am. Leather Chem. Assoc., 48, 735 (1953). H. J. C. Berendsen, J. Chem. fhys., 36, 3297 (1962). B. M. Fung and P. Trautmann, Biopolymers, 10, 391 (1971). R. E. Dehl and C. A. J. Hoeve, J . Chem. fhys., 50, 3245 (1969). C. Migchelsen and H. J. C. Berendsen, J . Chem. fhys., 59, 296 (1973). C. A. J. Hoeve and P. C. Lue, Biopolymers, 13, 1661 (1974). C. A. J. Hoeve and S. R. Kakivaya, J . fhys. Chem., 80, 745 (1976). I. D. Kuntz and W. Kauzmann, Adv. Protein Chem., 28, 239 (1974). G. E. Chapman, S. S. Danyluk, and K. A. Mclauchlan, Proc. R . SOC. London, Ser. 6, 178, 465 (1971). G. N. Ramachandran and R. S. Chandrasekharan, Biopolymers, 6, 1649 (1968). E. Swuki and R. D. B. Fraser in "PeptMes, Pobpepties and Proteins", E. R. Blout, F. A. Bovey, M. Goodman, and N. Lotan, Ed., Wiley, New York, N.Y., 1974, p 449. E. L. Andronikashvili, G. M. Mrevlishvili, G. Sh. Japaridze, V. M. Sokhadze, and K. A. Kvavadze, Biopolymers, 15, 1991 (1976). D.Eisenberg and W. Kauzmann, "The Structure and Properties of Water", Oxford University Press, New York, N.Y., 1969, Chapter 4. R. E. Dehl, Science, 170, 738 (1970). S. Nomura, A. Hiltner, J. B. Lando, and E. Baer, Biopolymers, 16, 231 (1977). 6.M. Fung, J. Witschel, and L. L. McAmis, Biopolymers, 13,1767 (1974). N. Davidson, "Statistical Mechanics", McGraw-Hill, New York, N.Y., 1962, Chapter 16. M. Sugisaki, H. Suga, and S. Seki, Bull. Chem. SOC.Jpn., 41, 2591 (1968). P. J. Flory, "Statistical Mechanics of Chain Molecules", Interscience, New York, N.Y., 1969, Chapter 7.
A General Parametrization Method for Configuration Interaction in Spectroscopy Dennis Caldwell Department of Chemistry, University of Utah, SaR Lake City, Utah 841 12 (Received October 31, 1977; Revised Manuscript Received March 6, 1978) Publication costs assisted by the University of Utah
From the basic SCF equations relations are developed for the off-diagonal elements of the hF matrix in terms of the diagonal elements. The Coulomb and overlap matrices are expressed as functions of fundamental atomic quantities and certain group parameters to be selected for the particular series of compounds under study. A separate treatment of excited states for derivatives is given in terms of CI matrix elements, which may either be used separately for small scale model calculations or in conjunction with the more detailed SCF treatment. Emphasis is placed on the interpretation of experimental data.
I. Introduction Semiempirical methods have a twofold purpose: (1)the prediction of complex molecular properties from relatively simple atomic parameters; and (2) the interpretation of molecular spectra through the introduction of models which retain the essential features of the more elaborate calculations. In the first part of this work we shall examine the basic equations of approximate SCF theory and show that in order to obtain a plausible relation between diagonal and off-diagonal h F matrix elements, the Mulliken approximation must be supplemented by additional assumptions. 0022-3654/76/2082-1663$0 1.OO/O
The method is designed for determining parameters which exactly fit the spectral data of a parent compound. In the second part the subject of derivatives is explored, whereby a distinction is made in the treatment of simple and composite molecules. Parameters are introduced which allow one to make a simplified CI treatment with localized orbitals on each chemical group. From the basic state functions of the substituent groups and the chromophore, along with charge transfer configurations, a theory is formulated which allows both for a simplification of the many configurations required for moderately large molecules, as well as a model in which the CI matrix is 0 1978 American
Chemical Society
1664
The Journal of Physical Chemistty, Vol. 82,No. 14, 1978
reduced to its lowest terms for analyzing the behavior of the series as a whole. In many cases the salient features of observed data can be treated by a method amenable to a desk calculator in conjunction with the conventional large format calculations. 11. General Theory We begin by deriving a semiempirical expression for the matrix elements of the Fock operator which specifically combines the effect of electron screening with the core term. By means of the Mulliken approximation for multicenter integrals the Hartree-Fock equations1 may be written
h:q= h g + Fzp’,J(pqlrs)- 1/2(prlqs)] = h g + / 4 ZP ’ r s [ S p $ r s ( Y p r + ~ p +s Y q r +
F
/2SpeY?s(Ypq
hP4
- 1
-
I2%,[h;p
+ YPS + Yqr + r*)l (1)
+ h,Fl-
‘12PpqYpq
which implies far too small a binding energy. A useful empirical relation which suggests itself from the above derivation is found to be hrp
= hs) -
‘12 ~
hpFq = x/zs,,
- QAYAA- B
p ~p p p
i-
Pqqhpql-
+
[ 1 + bo(% nP
2, = np ( 2 1 p ) 1 ’ 2
(3e)
formally writing for virtual orbital hF matrix elements
dP=
OYPP
and replacing y p qin the CI matrix element expressions by
Ypp
Ypq
=
=
l/2(1
[2/(r,, +
i-
0)Ypp
r,,)+
Rpql
(4a) (4b)
The parameter u may be related to the difference between the ground and excited state sums over the selfinteraction contributions to the two electron density matrix
AT,
-
= pn(2) - P g ( 2 )
where
g A Q ~(2a) ~~~
+ h,F) 1/2(Ppp
=
~ss)
where h# is the core term. This relation is not invariant to transformation and must be used only for the nonorthogonal atomic basis, where the approximation best holds. A literal application of the approximation leads essentially to the result F
S,,
1/2PPqYPq
(2b)
where ppp is the canonical bond order matrix, which is formally related to the symmetrically orthornormalized atomic orbitals2 through the transformation, p = S1/2p’S1/2. Although there is no guarantee that will be real, in virtually all cases the overlap integrals are 10.5,and this requirement is invariably met. In more extraordinary cases one must redefine the density matrix in terms of complex conjugate quantities such that the elements will be real. Since the phase factors of the orbitals are arbitrary, the transformation will be unique, provided the signs of the square roots are all chosen in the same way. The other quantities have their usual significance: h(O) is the local atomic Hamiltonian for orbital p , QA and are net atomic charges, and penetration integrals have been negle~ted.~ In general these relations will be useful for any situation, provided the parameters are suitably chosen. The h F matrix in the nonorthogonal basis is thereby determined in terms of the overlap matrix S , the Coulomb matrix y, and the diagonal matrix, h(O). One may then follow the convention of setting hi! = -Ap, y p p = I p - Ap, and using the Mataga retention for y .4 The Smatrix may be found either from Slater orbitafl or by assuming a simple exponential interpolative formula. The parametrization scheme is summarized in eq 3 where K is a scaling factor to describe different types of bonds, such as u and P. By adjusting K and the overlap parameters for each type of bond one can optimize this method for the analysis of ionization energies and permanent dipole moments. For transition energies two refinements are employed involving a y p pindependent of hb? and a diffuse orbital correction for allowed transitions, which is related to the increase in the values of two electron density matrix elements for atomic orbitals. This latter effect may be estimated by
6;
Either a general linear relation between Q and A&, may be introduced or else the connection can be made without the introduction of any new parameters by writing 0
= (AFmax
- A-F)/AP,ax
(5)
where Apmaxis the largest value attained in a given series of related molecules. This sets the virtual orbital and self-interaction energies equal to zero for strongly allowed transitions, following the trend observed in a b initio calculations on benzene, ethylene, etc., where highly diffuse orbitals are encountered with low hP4and y p p values. This method is best suited to a comparison of allowed singlet with forbidden triplet transitions. In treating singlets alone other elements of the two electron density matrix must be involved.
111. Localization Models In conjunction with the preceding semiempirical program it will be desirable to have a simplified computational method for assessing experimental trends analyzing the results of the more detailed calculations. This objective is attained by using localized orbitals. Let it be assumed that the molecular orbitals for a given molecule are localizable to particular diatomic u or P bonds and to conjugated aromatic systems. The required spectroscopic properties are most critically dependent upon the nature of the configuration interaction matrix, which may formally be constructed by means of localized orbitals without a previous SCF step. In many cases the group orbitals can be written down by symmetry in conjunction with, when appropriate, certain experimental quantities such as dipole moments. In order to illustrate the method in its most common application, it will be assumed that the chromophore is a P system which has been substituted either with a second s-containing group or one with orbitals in hypercon-
The Journal of Physical Chemistry, Vol. 82, No. 14, 1978 1665
Configuration Interaction in Spectroscopy
jugation with the chromophore. Let the chromophore orbitals be designated xb,xA,* and those of the substituent xB,, xB,*. This will involve four types of configurations in the construction of the CI matrix: $AiAj* = I x A ~ X A ~ * ~
(6a)
/xA~XB~*I
(6b)
IxB~~A~*I
(6c)
IXB~YB~*I
(6d)
$ A ~ B ~=* $ B . 1A .1*
=
$BiBj*=
A typical term in the CI matrix is given by ( + A i A j * I ~ I $ AiBj*) = h F A j * B j * [(Aj*Bi*IAiAi)
where zf,B,U) = eiBU)ciBu)is the appropriate geometric mean. The parameter ycc is found from the relation for the n-n* transition in ethylene:
Since /3 is the hFmatrix element between the second and third atoms of butadiene, a typical off-diagonal element of the CI matrix is
*
($ AB IHI+
- 2 ( A i A j * lAiBj*)]
= egj*
- €Ai -
[(Bj*Bj* IAi A,) -
2 ( A i B j * lAiBj*)]
where the second term in brackets vanishes for distinct groups, and the first is related to the Coulomb matrices yA,ye of the two groups. If the Coulomb contributions to the diagonal terms were neglected, the diagonalization of the CI matrix would recover the delocalized canonical Hartree-Fock orbitals. In principle all the parameters except those linking the two groups, hA,Blor h4*Bl*, can be found from data on the separate groups. In more complex cases the method can be supplemented by the prescriptions given in eq 3 and by the construction of the appropriate SCF matrices for fundamental heterocyclic and other low symmetry systems. For example, the benzene chromophore, which is given a detailed treatment in the next section, requires three parameters: h&cl, hbc,, and yclCl. The first may be found from the relation, hclcl = KIC, for an assumed value of K (ca. 2/3) and the second from
-€ion
= hFC,C,
+ hElcz
with a ZDO formalism being assumed which incorporates the neglected nonnearest neighbor matrix elements into the group parameters. Then yc may be found by fitting the data on the two lowest forbidden transitions. The pyridine chromophore requires three additional parameters, h&N1, hEN, and y N I N The first is found in analogy with hclcl, f and the second' may be determined by an iterative method which diagonalizes the SCF matrix, fitting the lowest experimental n-ionization energy to the appropriate eigenvalue. Then y N I N l may be found by fitting the lowest forbidden n-ir* transition energies to the average of the experimental values. As a simple example butadiene may be considered. Four experimental quantities are available: hElc1= -K IC along ) : :1~ from butadiene. with E*, and @) from ethylene and The invariance requirements on the trace of the 2 X 2 SCF matrix for the ethylene ir system and the determinant of the 2 X 2 matrix for the occupied ethylene orbitals in the composite molecule give ECC (TA
* + (-E::;')
IhF InB) =
' / 2 P ": =:$E[-
=
2h&,
-
(7a)
'"
(7b)
= -(TB
* IhF ITA *) =
-I/ 2 0
This gives
Since xA,* and xB,* have no atoms in common, the Coulombic terms vanish. In general all off-diagonal elements in this model will be simply related to hFmatrix elements. The diagonal matrix elements are given by ($AiBj*IHI$AiBj*)
AA *)
'/zP
-'lzP
E'
0
-'/zP
'l2P
0
E'
where E' = tCc*- (-@) - l/JyI3 + 7 1 4 + 7 2 3 + 7 2 4 ) and the yi. are all related to bond distances and ycc through the d a t a g a relation. The parameter p may also be fitted to the series of nonaromatic conjugated molecules related to butadiene, rather than through the use of (7b). In this model consistent errors inherent in semiempirical SCF theory can be minimized by directly using the experimental values of the fundamental chromophore absorption frequencies in the CI matrix for the derivatives, rather than by calculating them from atomic parameters. The literature contains many examples of localization method^^-^ which can be used to advantage in casting a detailed semiempirical method such as the one in the preceding section into a form compatible with the localized model described here.
IV. The Example of the Benzene Derivatives This case lends itself particularly well to illustrate the general method described in section 111, because the orbital coefficients and the configuration interaction coefficients are determined by symmetry. Accordingly one begins with the n system in which there are the six orbitals: $l $2
( ~ N T )+[$$ 2 + ~4 3 + $ 4 + $ s + ~~1 ( l o a ) =( 1 / ~ [ 2 + 4 +~z - $ 3 - 2 $ 4 - $ 5 + ~~1 =
(lob)
( 1 / 2 m 2 + $ 3 - $ 5 - 461 (ioc) = ( i / m w 1 - $ 2 - $ 3 + 2$4 - $ 5 - ~~1 $3
$4
=
(1/2~4 $~3 + 4 5 - $61 = ( l / n ) [ $ I - $2 + $3 - $4 + $5 -
(10d) (ioe)
+s =
$6
$61
(10f) (Throughout this section all molecular orbitals will be based on orthogonalized atomic orbitals.) The first three of these are occupied and the last three virtual. The orbital pairs, (+2, G3) and (q4,+J, are degenerate with energies, a + /3 and a - p, respectively. (Here the hpFq for nonnearest neighbors will be set equal to zero.) From symmetry considerations one is led to a study of
1666
Dennis Caldwell
The Journal of Physical Chemistry, Vol. 82, No. 14, 1978
the excitations from the El, (q2,q3)level into the Ezu(q4, $ 5 ) level, which gives rise to the four state functions: $1
=
(l/fl)[I$zF41+
1$3~slI
(Ila)
$2
=
(1/fl)[l$z~sl-
1$3F41]
(Ilb)
$3
=
(l/fl)[1$2F41-
1$3FslI
(IlC)
1
i
$A4
=
[$AT413
$AS
=
l$AFSl,
.**,
$F4
=
$FS
= l$FFSl
l$FF4I
=
I$ZTA*l,
.**)
$ZF* = 1 $ 2 T F * l
$3A*
=
1$3$A*I,
-**?
$3F*
=
fBi
= fa
fZA*
=
S=
=0
03.04
[E][:][]r]
= eo
The Hmatrix in eq 1 2 reduces to a 12 X 12, which in the case of CZusymmetry splits into two 6 X 6 matrices for X
a * ]
(13)
'/2b2,-l/2b3,-b4,
=-
fl
%b6]
[o, bz, b3, 0,- b 5 , -b6i
~ K z
b6')] ' I z
It will be seen that k 2B2
=
-k 1 B 1 -kz B 2 a 3
=
kz Bz with similar relations for the Pi. The four vectors displayed in eq 13 are accordingly in a one-to-one correspondence with the cyi and S is unitary.
fB4 r
0
i
a 1
-
f2B*
0 2 *03
where the remaining eight column vectors are an arbitrary set orthogonal to the first four, and (in row form)
B2
bl* = (Q1IhFI$**)
-
=
=0
This allows the transformation matrix S to be formed for the middle matrix in eq 12.
I$3$F*I
etc. L If the cAz, cBz, ... values are not too dissimilar for a polysubstituted ring, an illuminating simplication will occur by taking €Ai
0 1 *04
-l/Zb5,
The first 12 are donor and the second group acceptor state functions. These together with the basic four of eq 11 will comprise the basis for the 28 X 28 matrix of the general benzene derivative. Strong donors will be characterized by low values of diagonal elements EA47 etc. and large values of hx2, hi3, while strong acceptors will have low c28: and large hrA*,hiA*,etc. The benzene derivative matrix will have the form shown in eq 1 2 for donors and weak acceptors where the vectors cyi, Piare given in Tables I and 11. The quantities b, and b: are defined by IhFI$A)
=0
B1 =l/(flkl)[bi,
$2A*
bl = ( 6 1
01 ' 0 2
Fsl
I$ 3 F 4 l I (1ld) which have the symmetry representations, El,, BIU,and Bzu,respectively. Associated with the six substituents A, ...,F are six occupied qA,..., $F and six virtual orbitals qA*, ..., qF*,which lead to the 24 charge transfer configurations:
( l / f l )1$2 [
=
$4
and Y polarized transitions. In the latter case a3 * a 4 = 0 *a4 = 0 a2 '(113 = 0 ( ~ 1 ' 0 1 2= 0
1
H= 0 0
0
f3F
The Journal of Physical Chernistty, Vol. 82, No. 14, 1978 1667
Configuration Interaction in Spectroscopy
TABLE I: The Matrix Elements HA4,i,
E,-.%
Elu-Y
HA',^
Blu
TABLE 11: The Matrix ElementsH,A*,i, H 3 ~ * , i PI
B2u
P2
0 3
P4
2 A* A4 2B* B4 2c* c4 2D* D4 2E* E4 2F* F4 3A* A5 3B* B5 3c* c5 3D D5
*
3E
*
3F
*
E5 F5
TABLE 111: Intensities and Energies for the El, Band in Methyl Derivatives of Benzene
By subjecting H to the transformation
Mono2,3-Di1,4-Di2,3,5,6-TetraPentaHexa-
T= where
Ecalcd
Eexpt
fcalcd
faxpt
0.242 0.231 0.235 0.228 0.220 0.218
0.242 0.239 0.236 0.230 0.226 0.225
0.887 1.445 0,910 1.436 1.287 2.218
0.946 1.230 0.975 1.484 1.543 1.954
-l/2b6*]
and Kl*, K2* are identical in form with K1,K2,one obtains two independent matrices (eq 14a and 14b).
-
-
The transformed basis vectors have the form 1
while the eigenvectors of Hy, Hxare given by
The Journal of Physical Chemistry, Vol. 82, No. 14, 1978
1660
~
~=
cp) (1u-x0 $ElU.X + Cg' $ B , U + c$l$Xl e c 9 p x 2 i- C$&$X,* + cw;*$xx,* (15b) ZU
It then follows that the intensities are measured by the vectors
($,IPI(H); where
Y1
=-
a
6K1
X I = - fi b 2 b 2 + 16K1
P3b3
+ PSbS + P6b61
Yl*, Y2*are identical in form to Yl, Y2,except for a change in sign, and X1*, X2* correspond to XI, X 2 with the same signs. The quantities p; are given by P1 = (@A IpY P
1*
=
($1
IPY
I$1)
[$A*)
(17a) (17b)
and are found from experimental data using the methods of the preceding section. From the properties of the matrices in eq 14 it will happen that for small negative values of 6; and 6;' the intensity of the allowed El, band will be increased, as is shown by the results for the methyl derivatives in Table 111;when these values increase, an ultimate decrease due to mixing with the forbidden B1, or B2, transitions is found, as for example in aniline. In Table I11 the parameters have been chosen to give F,= 0.29 au and bi has been adjusted to give agreement for toluene in the energy of the El, band (bi= 0.026 au). Higher values of za could well be used with the bi adjusted accordingly, and the
Dennis Caldwell
results would be substantially the same. The main purpose of this example is to show that the simplified CI model has the capacity for delineating moderately intricate trends in the spectra of substituted polyacenes. A more detailed quantitative study will be presented in a forthcoming publication. V. Conclusions In this communication we have, working largely within the framework of conventional semiempirical theory, outlined a program for tailoring parameters to fit particular classes of chromophores, such as the polyacene series. These parameters have been reduced to their lowest terms in order to take into account the following: (1)adjustment of atomic ionization energies to the molecular environment; (2) determination of off-diagonal hF matrix elements in terms of short-range and long-range relationships, which depend on the atomic parameters and bond distances; (3) corrections for diffuse orbitals associated with strongly allowed T-T* transitions. The resulting equations may be compared to those of the Adams-Miller program.lOJ1 A localized CI treatment of substitution has been given which may either be used in conjunction with the SCF or as a separate program for dealing with molecules whose group orbitals have been obtained by more rudimentary methods such as symmetry. The parameters are designed with this specifically in mind, so that trends may be studied with the aid of small scale computations, either as an end in itself or as a preliminary to more detailed computations. Acknowledgment. The author thanks the National Science Foundation for their financial support through Grant No. CHE76 17836A01 and the National Institute of Health for their support through Grant No. GM12862-13. References and Notes (1) J. A. Pople and D. L. Beveridge, "Approximate Molecular Orbital Theory", McGraw-Hill, New York, N.Y., 1970. ( 2 ) P.-0. Lowdin, J . Chem. fhys., 18, 365 (1950). (3) M. Goeppert-Mayer and A L. Sklar, J . Chem. fhys., 6, 645 (1938). (4) N. Mataga and K. Nishimoto, Z . fhys. Cbem., 13, 140 (1957). ( 5 ) C. Edmiston and K. Reudenberg, Rev. M o d . Phys., 35, 457 (1963). (6) W. von Niessen, J . Chem. fbys., 56, 4290 (1972). (7) V. Magnasco and A. Perico, J . Chem. fhys., 47, 971 (1967). (8) K. Roby, Mol. fbys., 27, 81 (1974). (9) D. J. Caldwell and H. Eyring, "Localized Orbitals in Spectroscopy", Adv. Quantum Chem., in press. (10) 0. W. Adams and R. L. Miller, J . Am. Chem. SOC.,88, 404 (1966). (11) 0. W. Adams and R. L. Miller, Theor. Chim. Acfa(Berl.), 12, 151 (1968).
