Feb. 5, 1953
= 6.28318
- 3.28993'
- 6.28318 (1 - 1.0470F)'/*
(11)
The last term in equation (9) is zero when F = 0 and increases as F increases. Hence, for any value of F, equation (11) gives too high a value of B1, the error being zero when F = 0. Now, at a value of F = 0.85, equations (8) and (11) give values of Bt agreeing t o within 0.005, corresponding t o a variation of F a t this point of less than 0.001. Hence equation (11) was used for values of F from 0 to 0.85 and equation (8) for values from 0.86 to 1. The results are shown in Table I. The errors in Bt are less than those corresponding to a variation of 0.001 in the value of F.
TABLE I RUSULTSI N TABLEFOR INTERPRETING EXPERIMENTAL TERMS O F THE PARTICLE DIFFUSIONEQUATION F
Bf
597
II-ELECTRON SPECTRA
F
Bt
F
Bf
0 0 0.25 0.0623 0.50 0.301 0.01 0.00009 .26 .0678 .51 .316 .02 .00036 .27 .0736 .52 ,332 .03 .00076 -28 .0797 -53 .348 .04 .00141 .29 .0861 .54 .365 .05 .00219 .30 .0928 .55 ,382 .06 ,0032 .31 .56 .0998 .400 .07 .0044 .32 .I070 .57 .419 .08 .0057 .33 .I147 .58 .438 .09 .0073 .34 ,1226 ,59 .458 .10 .0091 .35 .1308 .60 .479
[CONTRIBUTION FROM THE
.ll .12 .13 .14 .I5 -16
.I7 .18 .19 .20 .21 .22 .23 .24
.0111 ,0132 ,0156 ,0183 ,0210 .0241 .0274 .0309 .0346 .0386 .0428 .0473 .0520 .0570
.36 .37
.
.38 .39 .40 .41 -42 .43 .44 .45 .46 .47 .48 .49
.1391 .1485 ,1577 ,167 ,177 ,188 ,199 .210 .222 ,234 ,246 .259 .273 .287
.61 .62 .63 .64 .65 .66 .67 .68 .69 .70 .71 .72 .73 .74
.86 1.468
.500 .522 .545 .569 .594 .620 .647 .675 ,703 ,734 .765 .798 .832 ,868
,87 .88 .89 ,90 .91 .92 .93 .94 .95 .96 .97
1.543 1.623 1.710 1.80 1.91 2.03 2.16 2.32 2.50 2.72 3.01 .98 3.41 .99 4.11
This work forms part of the research program of the Chemical Research Laboratory and is pub0.75 0.905 lished by permission of the Director of the Labora.76 ,944 tory. The author desires to express his indebtedness to .77 .985 .78 1.028 Dr. K. W. Pepper for his advice and encourage.79 1.073 ment, to Dr. T. R. E. Kressman and Dr. R. F. .80 1.120 Hudson for helpful discussions, to Mr. W. F. Wall .81 1.171 for carrying out some preliminary experimental .82 1.224 work and to Mr. F. W. J. Olver for advice on the .83 1.280 mathematical transformation of the particle diffu,84 1.340 sion equation, F
B1
.85 1.404
DEPARTMENT O F CHEMISTRY
TEDDINGTON, MIDDLESEX, ENGLAND
AND CHEMICAL
ENGINEERING, UNIVERSITY
OF \?IASHINGTON]
On the Use of Structures as an Aid in Understanding rI-Electron Spectra BY WILLIAMT. SIMPSON RECEIVED MAY27, 1952 The possibility of obtaining symmetry species from known transformation properties of the squares of trial wave functions is investigated. Application to structures follows from the association of structures with wave functions squared. Observed electronic term values are arranged in a diagonal matrix which is then transformed into non-diagonal form. Base vectors in the transformed cdrdinate system are interpreted with respect to structures. I t is pointed out that the energy matrix in non-diagonal form lends itself t o various applications.
Introduction The HLSP or valence bond method' provides a particular quantum mechanical basis for an understanding of the role played by structures (also called mesomeric forms, resonance forms, paper structures, etc.) but certain objections arise. In large molecules the wave functions which, according to the valence bond scheme, correspond to the principal or unexcited structures make only small contributions. Moreover, the valence bond scheme as usually employed involves the neglect of quantities which are certainly large and does not ordinarily take into account the so-called a-bonds. One wonders, in view of the practical uses to which structures can be put, if there is not some fundamental justification for an approach other than the valence bond method, which also involves the use of structures. In the next paragraphs several rather striking examples of the utility of structures will be cited. (1) Recent treatments have been given by D. P. Craig, Proc. Roy. SOC.(London). APOO, 27%(1050). aad h l . Simuuelta and V. Schomaker. J. Chcm. Phys., l S , 840 (1H51). (2) .A, Pullmuu, Doctoral 'lhwia, Unlv. of Paris, 1W.18.
It is fouiid that the 2600 A. band in benzene is in all probability AI, 3 Bzu8 and this is exactly what one expects if the transition is considered to take place between $, and where Itg
= 2-'h
($1
+ $d, It. = 2-'h
($1
-
$2)
and $12 has the transformation properties of one KekulC structure, &2 of the other.4 It is important to note that structures are taken as corresponding to $2's. The transition moment integral in the language of wave functions (squared) related to structures is
Sh
+ iI.e
dr =
51 [S$1'
+ 9dr
4
- S h24 d r l
that the electric moment implied by a distribution of charge transforming like one KekulC structure must be different from that implied for the other in order than the transition be allowed. That the electric moments are in fact equal may be seen in pictorial fashion from the Kekuld structures SO
(2) See for example F. hl. Oarforth, J . chirn. phrs., 46, 8 (1948) (4) This is demonstrated in Part I.
8
realized that the writing out of the conventional structures is an indispensable prerequisite to molecular orbital calculations. The approaches to the color of cyanine dyes by L. G. S. Brooker and $1 the color of triphenylmethane dyes by G. E. K. I t is convenient to take the center of the molecule to Branch make a most extensive and fruitful use of as origin. If para disubstitution is introduced structures, and in a way which need not be conR R sidered to be quantum mechanical in the sense of the HLSP method. It is the object of this paper to give a method for Ycombining quantal and group theoretical ideas with work with structures of an empirical nature (mesomerism). The result is a scheme which is not in1 I R R consistent with quantum mechanics but which is it is seen that the electric moments generated must quantum mechanical only by implication. The plan is to start with observed electronic necessarily be in opposite directions along the y axis5 The difference of the moments no longer energies for a given system, regarding them as vanishes and the perturbed 2600 K . transition diagonal elements of the energy matrix in a Heisenbecomes allowed ( y ) , which is required by more berg representation. Then instead of making the conventional arguments based on group theory.6 prohibitively difficult transformation to the SchroThis method of predicting polarization using struc- dinger representation (which is fully pictorial, with tures is of course not restricted to benzene but can base vectors the positions of all the particles) one be applied whenever it is possible to describe the makes a transformation to an intermediate represtates between which the transition takes place as sentation characterized by the property that the the sum and difference of two trial wave functions base vectors are associated with structures. This new representation is called the structure repreeach related to a structure. Another example of the application of structures sentation. It has certain of the qualities of the arises in a study of the triphenylmethane dyes.’ Schrodinger scheme in that to a degree there is a “shape” or configuration aspect; hence there is The general dye is greater opportunity for the exercise of quantum mechanical and chemical intuition than with the Heisenberg scheme. Matrix elements in the structure representation can be used to systematize data for families of related molecules and to act as a basis for predicting unknown matrix elements. The present paper is divided into two parts. k 3 The first is devoted to the relationship between where the numbers serve to indicate the position species of levels and structures and hence to polariof the charge. For example if -X is -NKz (and zations of bands. The second part deals with the matrix transformation theory and hence with the I
