DAVIDPETERS
3812 [CONTRIBUTION FROM THE
DEPARTMENT OF
CHEMISTRY, ROYAL HOLLOWAY COLLEGE, SURREY, ENGLAND]
VOl. 84 ENGELFIELD GREEN,
The Population Analysis of Valence Bond Wave Functions BY DAVIDPETERS RECEIVED DECEMBER 8, 1961 The population analysis of valence bond (v.b.) wave functions is shown to lead to a new way of understanding the role of the structures of valence bond theory. It also gives a detailed insight into the electronic organization in molecules as estimated by valence bond theory. Three examples, HI, LiH and the butadiene ?r electrons are discussed.
Introduction
correspond to orthogonal functions and so be mutually exclusive. The most recent and deOne important objective of theoretical chemistry tailed effort in this direction is Shull's work6 on the is the provision of simple pictures of the electron two electron homopolar bond. This approach distribution within molecules. Such pictures can departs rather far from the conventional ideas of never be precise, but they provide the only easy valency, and it now seems likely that no simple way of visualising the factors which govern molecu- and perfectly satisfactory solution to the nonlar structure. To this end, various quantities have orthogonality problem of valence bond theory will been defined' which are said to give such insight be found. Nevertheless, all the successful but into the electron organization within molecules. qualitative ideas of the valence bond theory of the These quantities are all derived from the concept structure of large molecules, due in the main to of atoms in molecules and the resulting idea of the PaulingI4 are so useful that some method of charge on an atom, or in a bond, in a molecule. analysing valence bond wave functions is desirable. The most thoroughgoing definitions of such The procedure suggested in this paper is a popuquantities are the population analysis formulas lation analysis method. This is a compromise proposed by Mulliken.2 These were originally between a formal analysis and the intuitive ideas defined to apply to the results of a one determi- of chemical valence theory. From a formal point nant molecular orbital (MO) wave function, but of view, i t has the disadvantage of starting out they have since been developed further by Karo3 from an arbitrary form of the wave function itself to deal with many determinant MO wave functions. and of the basis functions. Its advantages are Since it is now feasible to carry out calculations on first, its close connection with MO population a valence bond basis and since it is most helpful analysis and with chemical thinking in general, if both MO and valence bond wave functions can and second, the ease with which it may be exbe interpreted in the same way, we report in this tended to deal with large molecules. paper a valence bond wave function population The general approach to any kind of population analysis which parallels that of iLl0 wave functions analysis is simple. Given a 2n electron wave already in use. Previously, the general method of function, \k, for a molecule, this is a function of interpreting valence bond wave functions4 has been Gn spatial coordinates (XI, yl, z1, . . . . .zzn) and quite crude and very different from that used for 2n spin coordinates (al., . . . ( T Z ~If) . \k is normalhIO wave functions, so i t has been difficult to com- ized to unity, then the function pare the results given by the two kinds of wave p(x,y,z) = 2n f **qdx?,dy?,dzz. . .dzpn dut. .dum function. It turns out, furthermore, that the population analysis also leads to a helpful way of under= 2n f \JI*Odr?.. .dnndul (1) standing the role of the structures of valence bond theory. Many earlier interpretations of the sig- is an electron density function in real space, denificance of these structures have been unsatis- scribing the probability of finding an electron a t factory since they assumed that the wave functions the point x,y,z of real space. ( I t is important to of the structures are orthogonal. I n fact, the notice that much of the information in the wave structure wave functions are sometimes very far function (\k) is lost during this integration, so that from orthogonal and so one structure is in part p is much poorer in information than is \k. The second order density matrix, which has the discontained in another. This situation has been discussed a number of advantage of being much more complicated than p, is as far as one can go while retaining all the inand various attempts have been made to formation present in the wave function.) If \k recast valence bond theory in a form which makes is already expressed as, or can be transformed into, such concepts as the covalent and ionic structures sums and products of the AOs of the constituent atoms (x), then p can be put into the form of sums (1) C. A. Coulson, Proc. Roy. SOC.(London), 1698, 419 (1939). and products of AOs of the kind XaXa and XaXb. B. H. Chirgwin and C. A. Coulson, ibid.. Bola, 196 (1950). R. McWeeney, J . Chcrn. P h w . , 19, 1614 (1950); 20, 920 (1951). The electron density of the molecule, then, is a (2) R. S. Mulliken, ibid., 28, 1833, 1841, 2338, 2343 (1955). superposition of a number of simpler density func(3) A. M. Karo, ibid., Si, 182 (1969). tions, each of which is associated with a particular (4) L. Pauling, "The Nature of the Chemical Bond," Cornell UniA 0 or with a product of two different AOs. versity Press, Ithaca, New York, 1960. (5) J. C. Slater, J . Chcnt. Phys., 19, 220 (1951). R. ZIcWeeney, Now there is no unique way of expressing the Proc. Roy. SOC.(London), 427A, 288 (l954), a n d previous papers. molecular wave function in terms of the AOs of J. Braunstein and W. T. Simpson, J . Chcm. Phyr., 2 3 , 174 (195.5). R. S Berry, ibid., SO, 936 (1959). H.Shull, J . A m . Chcm. SOC.,84, the constituent atoms, but in practice one particular way is usually clearly more sensible than others. 1287 (1960). I
2 .o
2 .o
I .o
I .o
/
'\
p Q OO
x
x Fig. 1.-Structure populations, Mco and Mi,,overlap populations, moo (a,b) and mi, (a,b), and the total overlap population, m(a,b), in the HZlike case: &,b = 0.50: see eq. 12 and Table I.
It is also true that, in the molecule, it is convenient to work with one electron functions which differ somewhat from the AOs of the free atoms, but such modified AOs can always be expressed as linear combinations of the free atom AOs and the analysis completed, in principal a t least, in terms of these. It is natural, then, to think of these different contributions t o the total density function as charges, or populations, on an atom or in a bond. Further discussion of the general ideas underlying population analysis may be found in the earlier papers.'P2 Summary of MO Population Analysis.-From the point of view described in the preceding paragraphs, it is easily seen that the population analysis of MO and valence bond wave functions are closely related and, before discussing the valence bond analysis, i t is helpful to summarize the &IO treatment in a form which emphasizes the parallels. The analysis of MO wave functions in which configuration interaction is included divides into three steps. We consider in the usual way a 2n electron ground state wave function ( ~ M o ) =
Cy
%$3
0.5
0.5
*MO
3513
A FUNCTIONS ~
POPULATION ANALYSISOF VALENCEBONDW
Oct. 20, 1962
f zp$a
(2)
B
whefe $0 = . . . . . .qin&1 is the first approximation to the state wave function. The remaining J. are formed by raising one or more electrons to excited MOs. As an example, suppose that one electron is raised from the ground state MO +i to the excited M O 4 4 , then $ p = I+a$a. . .+i. . . +,,&. . + q l . The functions $o,etc., and the MOs are normalized to unity so thatBa
.
(6) (a) To simplify the treatment, we have supposed that all the 4.3 are orthogonal to and to each other. From e numeiical point of view, more rapid convergence is obtained if the &p are selected from the more general and flexible set of simple functions which nre not orthogonal to +a. Then configuration overlap populations arise which
Fig. 2.-Structure populations, M,,and Mi,, overlap populations, m,,(a,b) and mi,(a,b) and the total overlap population, m(a,b) in the HZlike case: &,b = 0.25; see eq. 12 and Table I.
1 = a 2 + p ... (3) The density function which results from ( 2 ) is p ( x . y , z ) = 2na2f$o* $Odn. . .d7zn dul 2n p 2f $a*$@ d T p . . . d n n d u l
+
=
+ 2@b*
Cy2(2a*$s
B
4b
.. .) +
pz(2&*&. . . )
(41
13
plus, in some cases, products of the kind +j*+q which individually integrate to zero. It is important to notice these latter terms which, although they make no contribution to the final numbers, do show that the electron density function itself is not quite so simple as a direct superposition of the individual density functions of the occupied MOs. Apart from these terms, p does appear as a sum of terms each of which is connected with a particular configuration. We call the quantities 2na2, etc., the configuration populations6band denote them by N,, etc. The second step in the analysis of MO wave functions follows immediately from the second form of writing in equation 4. The configuration population 2na2, for example, is divided up among the os, 2a2 electrons in $a, 2a2 in @q, and so on. The third step is the division of the population of an M O among the contributing AOs via the equations 1= f =
d7 =
Cark2.fXrk*Xrk
f
(
zrk C.rkxrk
d7. .
)*( p
0
1
.. +
2Srk,slCrkCdf-
xIk*xII srk,d
x
d7.
y
. ..
(5)
Here, the rth A 0 of atom k is denoted Xrk. From this point of view, the various atom and bond populations are defined in the usual way.a,a resemble the structure overlap populations of the valence bond theory which are discussed later. (b) Karo; called these quantities N ( & . The notbtion used in the present paper is that N or n denote a quantity obtained from a MO wave function while M or m denote that from a valence bond wave function.
DAVIDPETERS
3814
Valence Bond Population Analysis.-Having outlined briefly the population analysis of M O wave functions, we return to valence bond wave functions. Now the analysis follows very similar lines. There are two steps, the first of which is analogous to the division of the electrons among the M O configurations. We write the normalized 2n electron ground state valence bond wave function as *VB
= A h
f p$Jp . . .
(6)
*VB
Vol. 54
A\I.oovalent
=
f M\I.lomc
+
+
A(n(l)b(2) u(2)b(1))(2 2.%,bZ)-1/3 p(n(lia(2) h [ l ) h ( 2 ) ) ( 2 ".%'ab2
+
+
+ )-'I2
(10)
To be quite specific] we suppose that we are using M'einbaum's HZwave function] with free atom hydrogen AOs in both the covalent and ionic wave f ~ n c t i o n s . ~These are written a ( l ) , etc., and S a b = fa(l)b(l)dxldyldz,. Equation 8 is now 1 = A*
+ + -"ApSA, p2
(11)
and the gross structure population of the covalent structure Mco is (2X2 2XpSX,) and of the ionic structure ilf,,, ( 2 ~ * 2XpSA,). Numerical values for these quantities are given in the last column of Table 11. I n the extreme right hand column of Table I11 and in Table V are given the gross structure populations of the valence bond wave functions of p ( S , y , Z ) = 2n f*VB**VB dTz. . .d~zn.dui LiHs and of the x electron system of b ~ t a d i e n e . ~ = 2nAZf $X*$X d T 2 . . . d n n dgi We take up the discussion of these results later in this paper. 2np2f $ p * $ p d r z . . . d n n dai. . . HurleylO has expressed similar ideas in a rather 4nAp f +x*\I.~ d r 2 . . . d ~ ~ , d .u. ~ .( 7 ) more cbmplex mathematical format. Hurley's The function p is analogous to that from the M O comment that the gross structure populations wave function (equation 4) and must be interpreted provide a rigorous definition of the weights of the in the same way. If \EVB itself and all the $A are valence bond structures in a valence bond wave normalized to unity, we have function is formally correct, but i t is important to notice that this is not the only possible definition. 1 = x2 + p 2 . . . + 2 A p S x p . . . (8) Others in which the structure overlap populations where SA, = f $A*$pdrl.. .dT2n. The important are not divided up equally between the contributdifference from the simplest &IO case (equation 3) ing structures are possible. Hijikatalla has used is that SA, is not now zero. 1T-e call the quantities the idea to discuss the results of a calculation on FP 4nXpSAp the structure ocerlap populations, mp,A. and Brownllb has used i t t o discuss a number of These must be distinguished clearly from the over- LiH wave functions. lap populations between two AOs since the latter The second step, analogous to steps 2 and 3 of are localized in regions of real space while the the 310 treatment, is the division of the gross structure overlap populations are not so localized. structure populations among the AOs and the bond We call the quantities 2nh2, etc., the net structure regions. The general formulas are complicated, populatiow. These latter numbers have, in the and i t is convenient to set out first the equations for past, frequently been used to describe or define the simplest case of H2. Starting with the gross the amount of covalentness and of ionic character of structure population of the covalent structure of the valence bond wave function, the non-zero overlap H P ,which is 2(X2 XpSx,), this may be written out of the structure wave function being ignored. These in detail as numbers do not add to 2n, the total number of f (U'b' + U 2 b ' ) ' ( 2 f 2Stab2)-' d71d72 f electrons, since the overlap terms remain to be in- h f c o = 2[x2 A p f (U'b' + azb')(a'a2 + b'b2)(2 -f 2 S a b 2 ) - ' d T l d T 2 1 cluded. Just how these overlap terms are to be = %[A2 f (U'U' + b'b' + 2 S , b a 1 b ' ) ( 2 + 2 S a b 2 ) - ' d T 1 + divided up among the two structures which are A p S ( U ' U ' S , b + b'b'&b + 2 U ' b ' ) ( 2 2&bz)-'dTi connected with each one is in general arbitrary, = 2[A2(1 1 2Sab2)(2 2sab2)-' although in special cases, such as the two KekulC h(Sab Sab 2 S s b ) x (2 2&b2)-'] (12) structures of benzene, they must clearly be divided evenly. The same situation arises2 with the Then the net population on each atom in the covaoverlap terms of two AOs in an &IO. If we decide, lent structure, m,,(a) is 2(X2 XpSab2)(2 2The overlap population in the covalent for the present, to always divide the structure &b2)-'. overlap populations equally between the two structure, nz,,(a,b) is 2(2X2sab2 2XpSab2)(2 structures, we may then define a gross structure 2S,b2)-I. The gross population of one atom in the covalent structure] Mco(a),is (A2 XpSi,). These population, -ITA,for structure X, by definitions are consistent with those used for XI0 M A = 2 n ~ 2+ 2n x,~.sA,. (9) wave functions. In Table I are set out these quan,(#A) tities and the corresponding ones for the ionic These quantities, d l x , now add to 272, the total structures, together with certain total populations number of electrons, They may be thought of as (7) S. Weinbaum, J . Chem. Phys., 1, 593 (1933). the weights of the structures in the final wave (8) J. Miller, L Friedmann and F. A. hlatsen, Bull. Amev. P h y s . function. Reu., Ser. 11, 1, 90 (1956). (9) J. Fain and F. A. Matsen, J . Chem. Phys., 26, 376 (1957). To exemplify the idea of a structure population, Roy. SOL.(London), (10) A. C . Hurley, ibid., 28, 539 (1958); PYOC. we take the Hz molecule whose ground state va- MA, 119 (1958). lence bond wave function is, apart from a singlet spin (11) (a) K. Hijikata, J . Chem. P h y s . , 34, 230 (1Y61); (b) J. C . Brown, i b i d . , in the press. function]written as where $A, etc., are the conventional bond eigenfunctions, themselves usually linear combinations of determinants which are eigenfunctions of S, and S2 which correspond to different ways of joining together the 4 0 s of the atoms by single bonds. The electron density function in real space is now
+
+
+
+
+
+ + +
+
+
+ + +
+
+
+
+
+
Oct. 20, 1962
POPULATION ANALYSIS 01:VALENCEBONDWAVEFUNCTIO~VS
defined below. This table is immediately applicable to any problem which can be simplified until i t resembles the HB case. The figures illustrate the numerical values of the various populations for the two representative cases, S a b = 0.25 and 0.50. The numerical values for H2 itself, using Weinbaum's wave function,6 are given in Table 11. TABLE I1 POPULATION ANALYSIS OF WEISBAUM'S HIWAVEFUNCTION^ Population
struct ure
Net atomic mh(a) =
Overlap
mxb)
mx(a,b)
0.576 0 588 Covalent .068 ,124 Ionic .712 Total ,644 a Ref. 7, free atom a.0.s for both
Gross atomic ,Mx(a) = MX(b)
Total structure
0 .S i 0 0.130 1,000 and $,,.
1.740 0.260 2.000
$co
M X
With the general valence bond wave function, the structure populations become rather complicated if written as functions of the quantities defined below,12 so i t is easier if the equations analogous to (12) are developed for each case individually. We have, in fact, to transform equation 9 into an equation involving only AOs. The resulting equations can always be written in the form
3Sl5
The term 'partial' refers to a quantity of one structure, and the term 'total' implies that summation over all the structures has been carried out. The term 'net' implies that the overlap contributions have been omitted, and 'gross' that they have been included. Overlap Populations.-Partial overlap population between AOs Xrk and xSl in structure X = mX(rk,s1) = 4ndrk,slX. Partial overlap population between atoms k and l in structure X = mx(k,l) = mX(rh,sl). Total overlap population between AOs r,s Xrk
and xSi = m(rkS1) =
mx(rk,sl). Total overlap x population between atoms k and 1 = m(k,l) = x mh(k,l) = m(rk,s,). Total overlap population = r,s
m(k,l).
m = k,l
Atom Populations.-Partial net population of A 0 Xrk in structure X = mA(rk) = 2ndrkA. Partial net population of atom k in structure X = mA(k) = mi(rk). Total net population of Xrk = m(rk) = r
x =
mh(rk). Total net population of atom k m(rk)
=
x in structure X
m(k)
mx(k). Partial gross population of
r Xrk
=
=
+
Mx(rk) = 2n(drkA
drk,$iX), SI
In the double summation, ?'k = s1 is omitted. There are some obvious differences between this equation and the corresponding MO equation, although the latter can always be put into this form. I n practice, the derivation of (13) is straightforward providing the normalization conditions are dealt with carefully. This equation 13 can now be taken as the starting point in the definition of the various simple quantities analogous to those used for MO wave functions. Some of these are not required immediately, but they are included for completeness and to standardize the notation. To summarize the notation, k and 1 label atoms, r and s label AOs. Structure labels, written as subscripts, are X and 1.1. M always labels a gross population, either of an atom or of a structure and m a net atomic population or an overlap population, either between two AOs, two atoms or two structures. (12) Both Loadin's and Hurley'o have given formal equations for doing this.
(13) P 0.Lowdin, P h y s . Rev., 97, 1474 (19.55).
rk # SI. Partial gross population of atom k in structure X = Mx(k) = Mx(rk). Total gross population
&f(rk)
.i!,!fh(rk).Total x gross population of atom k = M(k) = XX(k). Of Xrk =
=
A
Total number of electrons
=
2n
M(k)
= k
Structure Populations.-Overlap population between structures X and 1.1 = mi,, = 4nX~Sx,. Net population of structure = 2nX2. Gross population of structure x = -4fx = 2 n ( ~ 2
+
Xpsx,)
=
2 n ( x drk' rk
ber of electrons
=
+
2n
=
r(#M
drk,slX) Total numrk # sl
x
MA.
Some of these populations have been derived for a LiH wave function' and these are reported in Tables I11 and IV. A complete valence bond wave function for the x electrons of butadiene has been reporteds and the derived gross structure populations are reported in Table V
DAVIDPETERS
3816
Vol. 84
TABLE 111 GROSSPOPULATIONS OF LITHIUMHYDRIDE5'* MX(2sLi) MX(ki) W.(W 0.492 0.020 0.654
MX(1SLi)
1.166 0.5i6
.398
M A W
MA
1.678
2,332
,025
.220
.330
0.820
1.152
,041
.028
.329
.467
0.796
- .055
-
,038
-
,064
-
,024
-
,020
0
-
.018
-
,092
-
,108
-0.020
- .020
-
,108
-
,128
-
-
-
,034
-
,040
0
.014
.00i
2.001 0.496 .234 M(H) M .L i ). = 2.i31 2n = 4.00 . . = 1.268 In the structure formulas, the (Is)* electrons of Li are omitted. * The valence bond wave function from which these results were obtained is reported in ref. 8. TABLE
Iv5
OVERLAP POPULATIONS OF LITHIUM HYDRIDE Structure
mA(lSLi, ISH)
mA(ZSL.i, ISH)
mX(zPLi, ISH)
mA
Li-H 2s 1s Li-€I 2p Is Li+ H -
-0.020
+0.308
$0,040
+0.328
-
+
+
+
(Is12
-
Li- H + (2s)* Li- €I+
(ZS)(2P) Li- H + (2PY
,004
,052
.204
,252
TABLE V" GROSSSTRUCTURE POPULATIONS O F A BUTADIEXE 7r ELECTRON WAVEFUNCTION C-C C-C 2 . 152
+ -
- + - +
Structure
CCCC
CCCC
W
+
,084
0
-
,036
0
- ,016
.008
+
,056
0 -0.024
+
,132
-
,036
-
,040
0 0 - .012 - ,012 7M(f'~i, ISH) -0.032 +0.392 f .264 m = +0.624 a See footnote to Tables 111.
Discussion The results reported in the tables provide some further insight into the behavior of valence bond wave functions, but they emphasize how complicated these wave functions are when examined quantitatively. I n Hz (Table 11) the structure populations are remarkable for the large size of the ionic population, M i o = 0.26. The numbers usually quoted4 as the relative weights of the ionic and covalent forms are 0.08 and 1.92, respectively. These are simply X2 and p 2 (renormalized to unity) of equation 10, the net structure populations. I n fact, Hz is a particularly bad case in which to ignore the overlap contribution since the covalent and ionic HeitlerLondon wave functions are much more nearly identical than orthogonal (the overlap integral between the normalized structure wave functions is 0.929). It is interesting to notice that, had we decided to divide up the overlap contribution in the proportion of X2:p2, we would also have obtained the populations 1.92 and 0.08. So the actual decision to divide up the structure overlap population equally leads directly to the large ionic population. There seems no reason, other than familiarity, to prefer the older numbers, but i t is helpful to bear in mind the origin of the large ionic populations. The overlap population between the two atoms in Hz in the ionic structure is 0.124 as compared
+-
n Structure l'updatiuii
-+
C C-C C C-C C C C-C C C 0.284 0.571 0.681
-
+ + - - +
CCCC
u
CCCC
Population 0.120 0.089 0.034 0.032 The wave function is recorded in ref. 9. The u electron lines have been omitted from the structural formulas. All structures with a population of less than 0.02 have been neglected. The listed structures account for 3.96 electrons.
with 0.585 for the same quantity in the covalent structure. If the overlap population is proportional to the binding, this means that some 20% of the total binding occurs in the ionic structure. This again is large and is a direct result of the equal division of the structure overlap population discussed in the last paragraph. The total overlap population in both structures is 0.712 which is close to that2 found with a one determinant M O wave function, 0,858. That the total overlap population in the valence bond wave function for H2 should be smaller than that for the one determinant MO wave function is to be expected. The valence bond wave function is identical with a configuration interacted MO wave function in which only the configurations which arise froni Is AOs on hydrogen are used. The latter wave function includes, with small weight, the configuration made from two electrons in the antibonding M O and this MO has a negative overlap population between the nos. This will reduce the total overlap population in the configuration interacted MO wave function compared with that found with a single determinant, in agreement with the results from the valence bond wave function. The two figures show how the various populations depend on the value of X and ~ . in l the H2 like situation. The gross structure populations, M,, and &Ila,are close to linear in X, becoming more so as the value of &b increases. For the usual values of S a b in a two electron bond, it is better to think of 1x1 and 1 ~ as1 guides to the structure populations, rather than the conventional X2 and p 2 . It is also noticeable from the figures that the structure
Oct. 20, 1962
ADDUCTS OF PI-IEKOL WITII NITROGE;.~ AND OXYGENDOSORS
3S17
population, M x , may be quite small (10 to 20a/0), Table III, show the expected 2.00 electrons in the but some 20 to 50% of the total binding may occur lithium Is AOs. The lithium atom as a whole, in this structure, judging from the size of the however, has lost 0.27 electrons to the hydrogen overlap population, mx(a,b), in it. Furthermore, atom. This over-all loss is made up of a loss of the total overlap population, m(a,b), is rather in- 0.50 electron from the 2sAO and a gain of 0.23 sensitive to XI particularly in the range X = 0.3 to electron by the 2puAO. More than this i t is impossible to say without making further as0.7. The LiH results (Tables I11 and IV) show nega- sumptions, since the amounts of promotion and tive gross populations, MA, for the three Li-H+ charge transfer are formally inseparable. It has structures. These are quite small, but far from been suggested2 that charge transfer be thought of negligible and their existence may be due either as involving only the 2puA0, and if this is so, we to the inadequacies of the wave functions them- can describe the lithium atom as 50% promoted to 2 s ) configuration. This agrees well selves or to the approximate nature of the popula- the ( 1 ~ ) ~ ( (2p) tion analysis formulas. A similar, but rather with Karo's results3 from the population analysis smaller, effect is found in the MO population of the LiH MO wave function in which configuration interaction is included. analysis.2 An interesting point about the LiH results is The butadiene results (Table V) are rather as their similarity to those of HZ, particularly in the expected on classical ideas of resonance between gross structure populations. In both cases, the valence bond structures. After the non-resonating covalent structure accounts for 87% of the total structures, the long bonded structure is the most population. Conventional ideas would say that important contributor to the ground state wave the ionic structures are more important in LiH function and this conclusion differs from that of which has a large dipole moment and ionic crystal. l 4 Berry.5 He rejected the wave function used here From the various overlap populations in LiH on the grounds that the ionic structures are under(Table IV), i t seems that a substantial amount of weighted in it, and this is a reasonable point of the binding occurs in the ionic structures. The view since the basis functions of the 2p, AOs were total overlap population in the molecule, 0.624, is represented by Slater AOs. Perhaps i t would be quite similar to that found in other small mole- valuable to repeat the calculations with SCF AOs cules2and agrees with that (0.741) for LiH from the for the 2p, basis functions, although there are some population analysis15of a MO wave function. The serious questions as to the behavior of the u elecgross atomic populations also agree rather well trons in butadiene. For the present, i t seems sufwith those found from the MO wave f ~ n c t i o n , ' ~ficient to say that the calculation based on Slater and this suggests that despite the imperfections of AOs does not contradict the classical ideas of the population analysis method, there is significant resonance in butadiene. agreement in its results for both MO and valence Acknowledgments.-This work was carried out bond wave functions. while the author was on leave of absence a t the Table I V reveals the expected small repulsions between the lithium 1s electrons and that of the Department of Molecular Structure and Spectra, hydrogen atom by the small negative values of University of Chicago. The author is indebted the appropriate overlap populations. The total to Professors R. S. Mulliken and F. A. Matsen gross atomic populations, given in the last row of for bringing the idea of structure populations to his notice and for much valuable advice during the (14) R. P. Hurst, 5. Miller a n d F. A. Matsen, J . Chem. Phys., 26, course of the work. Some comments from a ref1082 (1957). eree were also very helpful. (IS) S Fraga and R J. R a n d , ;bid., 34, 727 (1961).
[CONTRIBUTION FROM THE
WM. A. NOYES LABORATORY, UNIVERSITY
O F ILLINOIS, URBANA, ILLINOIS]
The Validity of Frequency Shift-Enthalpy Correlations. I. Adducts of Phenol with Nitrogen and Oxygen Donors BY MELVIND.
JOESTEN' AND
RUSSELLS. DRAGO
RECEIVED FEBRUARY 9, 1962 The correlation between frequency shift and enthalpy is tested by examining thermodynamic and infrared data for the interaction of phenol with thirty-three bases in carbon tetrachloride. Contrary to literature reports a plot of enthalpy v s . AVO-R is linear for phenol adducts. The relationship holds within f 0 . 5 kcal./rnole. The general equation derived for pre0.63. dicting enthalpy values for phenol-base association from frequency shift data is -AH (kcal./mole) = 0.016 APO-H The log K vs. AVO-E relationship is found to be limited to similar bases where no unusual entropy effects are observed and where the entropy change is a linear function of the enthalpy.
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Introduction I n 1937 Badger and Bauer2 proposed that the shift in the infrared stretching frequency of a group, (1) U. s. R u b b e r Fellow, 1961-1962. Abstracted in part from the Ph.D. thesis of M. D. Joesten, University of Illinois (1962).
X-H, upon complexation to a base was linearly related t o the enthalpy for hydrogen bond formation. Until recent Years, there has been little reliable enthalpy work available to test this pro(2) R. M. Badger a n d S. H. Bauer, J . Chem. Phys.. I , 839 (1937).
