Calculation of the electronic spectrum of formaldehyde

Volume 8, number 1

CALCULATION

CHEMICAL

OF

THE

S. D. PEYERIMHOFF,

PHYSICS

ELECTRQNIC

LETTERS

SPECTRUM

R. J. BUENKERt,

Institzcf ftir phJsikalische

Calculations

Gdenberg

of the electronic

1971

FORMALDEHYDE

and H. HSUS

Universitiit,

1970

for the ground and excited states of formaldehyde

description

OF

W. E.KAMMER

Chemie, Johannes 65 Maim, Germany

Received 4 November

a theoretical

I January

are carried

spectrum of this system.

out in order to obtain

The method

employed

combines

the SCF and CI techniques,

and is successful in obt$ning agreement with the experimental transition energies to within 10.2 - 0.3 eV. In particular the C state at 8.0 eV above the grcund state % is found to result from an n -3pb2 excitation, not the B -v* intravalence species previously believed; this result is shown to he quite analogous to that found in the study of the ethylene V *N transition.

1. INTRODUCTION Recently an extensive series of ab initio SCF and CI calculations for the formaldehyde molecule has been reported [l] t which investigates the structure of this system in ground and excited states. The method employed involved a combination of the SCF and CI techniques and was successful in obtaining transition energies to within 0.2 eV of their respective experimental values. The present study is an extension of this work designed to take account of the Rydberg excited states of H2C0, and also to allow for an improved CI treatment. The basis set for this work contains the same valence AO’s as in the previous calculations but also includes one-component s, px, py and pz diffuse gaussian functions at both carbon and oxygen; the new basis consists then of 68 gaussian (lobe) functions, contracted into 30 groups, eight more than used in 1. The exponent ac employed for the diffuse carbon functions has been chosen to be 0.02 (for both s and p), the optimum value found for this quantity in a series of ethylene calculations [Z]. The exponent a0 for the corresponding oxygen functions has been taken to be 0.05, qn the basis of additional optimization calculations carried out for H3CO. In each case the lobe separition R for the ditfuse p functions was determined according to the relation: R = 0.03 a -112, as discussed elsewhere [3,4]. $ Preseit address: Department of Chemistry, University of Nebraska, Lincoln, Nebraska 68508, USA. _? Hereafter referred to as I. . :,, : .

The ground state equilibrium geometry has been employed exclusively for the present calculations since the main interest in this work lies in the determination of the vertical transition energies for this system. The equilibrium geometries of the excited Rydberg species are not expected to be greatly different from that of the ground state; the potential curves for the ground and certain H2CO excited (valence) states have already been discussed in I. Total and orbital energies resulting from the ground sfate SCF calculation are given in table 1, and it .can be seen therein that addition of the difuse gaussian functiors to the basis set lowers

Table L

Total energy ET and orbital energies (in hartree) the ground state of W&O

--

for

.

la1

-20.5833

6%

0.0267

2%

-11.3559

2bx

0.0398

3=1

-1.4261

3b2

0 -0438

4%

-0.8675

?a1

0.0506

1%

-0.7031

3%

0 .LL73 (Try

Sal

-0.6389(u)

6%

O-L831

lb1

-0.5318

2%

-0.4403 (n)

(5)

4b2

0.1944

4=1 981‘

0.1972 0.2219. _

i+ =.-113.81286 129 - .I

\hmic

8. number 1

CHEMICALPH&ICS

LETTERS.

: the ground state total energy by only 0.0034

2. OPEN-SHELL

h&tree, in agreement with previous experience [2,5]. The eigenvalu,es for corresponding &IO*s .jn the two.calculations agree in general to within 0.003 - 0.004 hartree, the lower set resulting from the larger basis. Examination of the eigenvectdrs for the virtual MO’s finds’that the four most stable of these species are quite diffuse in character. In each case the carbon 3s and 3p AO’s make by far the largest contribution and this behavior is consistent with the fact that the electric dipole of H2CO is negative at the oxygen end. The next most stable MO (301) is mostly valence in character and corresponds to the 2~7;* species; it is fpllowed in the energy order by four MO’s which are composed in the main of the oxygen diffiise functions. A-similar calculation has been reported by Whitten and Hackmeyer [6], with their basis set lacking 3pu (al) functions on both b and C and a 3pn (b2) species on carbon; as expected, the occupied orbital energies of this calculation agree very closely with those in table 1 but certain of the virtual orbital energies are much different, because of the aforementioned distinctions in the basis set.

SCF energies gsCF (h.nrtrees) and excitation energies

the inherent correlation

error

in the SCF tech-

nique for ground and excited states. A series of open-shell SCF calculations has thus been carried out for the excited states of formaldehyde and the resulting total energies are reported in table 2. These calculations are carried out using the Roothaan scheme [?] and the states considered correspond to all possible single excitations (relative to the ground state)

Table 3 AE (ev) from the ground state for several excited states of H2CO AE

%CF -113.6129

0.0

-113.1301

2.25

(n *TIT)

-113.7164

2.62

@ + n':)

-113.6595

4.17

lAl

Ground state Zb2 -2’1)1

3

lA2

2b2 -22b1

4* 5*

3A1 3B2

lb1 -2b1 2b2--ia1

(n '3s~)

-113.5913

6.03

6

IB2

Zb2 - 6al

(n - 3sc)

-113.5898

6.07

7+

3Al

2b2 _ 3b2

(n - 3pnC)

-113.5561

6.93

a*

3%

5al -+2bl

(O-LHi)

-113.5573

6.95

9

lB1

5q '2%

(u-8+)

-113.5239

7.86 9 A2

(n +x”)

10

3*1

lb1 -681

(Tr* 3s($

-113.4813

11

lBl

lb, d6al

(r -3q$

-113.4796

(7r+3pnC)

-113.4452 ':

12

3412

lb1 -'3b2

13

'A2 %

lb1 -3b2

(T-L3pnC)

5a1 '6nl

&39C)

14

3”2

5al - 3b2

(r’3pnb)

,-

Q32

5al *3b2

Q“'3Pw)

:

?B2

Ion.

#I ‘00)

15 16

. 33'0

It has been shown [ 1,6] that a limited CI calculation for the excited states of H2CO which is based exclusively on the ground state MO’s greatly overestimates transition energies, unless .one employs a very Isirge-scale treatment. A potentially simpler method of obtaining good transition energies, which does not require the solution of high-order secular equations, combines the SCF and CI techniques. In this procedure, referred to in I as‘the CI(PCM0) method, the SCF MO’s of a given parent configuration are employed as a basis for its own CI expansion and those of other states of the same spin and spatial symmetry; as discussed in I the role of the CI calculatioc in this method is merely to balance

3h.,

1* -2*

-.17

CALCULATIONS

Excitation,

State

Calc:No.

1&lnuary 1971

._-_

-il3.4440

IO.02

-113..4039

11113

-113.3765

,.

.lZ.iIS .-

-113.4649

.-

.

9.47

.

.

:. . .

._ :-

12.04

-113,9+i?

.. .

.’ :

’

9.07

10.00

,.

:-.I

._

_.

,’

Volume 8. number 1

CHEWCALPHYSICSLETTERS

from the 5al (c), lb1 (v) and 2b2 (n) MO’s respectively, to the most stable MO possible in each of the al, bI and b2 irreducible,representaiions; in addition, an SCF calculation for the 2B2 round state of H9C!O+ has been carried out. No !zAl excited states corresponding to single excitations have been obtained, since such solutions are not accessible by means of Roothaan’s method. The character of the upper orbital in these open-shell wavefunctions generally reflects the ordering of virtual MO’s found in connection with the ground state SCF solution, but not in the case of the s*(2bl) orbital, which is invariably predominantly valence in character in the excited state calculations in contrast to its diffuse nature calculated in the electronic field of the ground state. These results are completely analogous to those found for ethylene in previous calculations [2]. In all cases for which comparison is possible the calculated SCF transition energies are substantially lower (by around 1.0 eV) than the corresponding experimental values, in agreement with considerations of the unequal correlation energies of the ground and excited states. The results for the positive ion are of particular interest; in this case the transition energy (I.P.) is underestimated by 1.4 eV, a relatively large discrepancy, occurring despite the fact that the Koopmans theorem I.P. (11.99 eV) considerably overestimates the experimental result. The aforementioned ethylene calculation, on the other hand, has found quite good agreement between experimental and Koopmans theorem I.P’s, while again underestimatxhg it by 1.5 eV in the SCF treatment; the indication is thus that the SCF method leads to much more consistent. ionization potential results than does Kocpmans theorem. Such a result cannot be expected to obtain in general, however; unless the A0 basis emplcyed is flexible enough to allow for the necessary contraction of the core of the ion relative to that of the neutral ground state. 3. CONFIGURATION INTERACTION:

CI(PCM0)

Because major interest lies in the most stable of the excited states, Cf calculations have been carried out only for the MO’s of these species. In accordance with previous experience it was decided to use the MO’s of a given triplet state SCF. solution for ,bo+b its own CI treatment and @at of the corresponding singlet; The SCF ,calculations whose-resultipg MO’s are employed z&i

basis for a CI treatment are shown with an asterisk in table 2 and are seen to be six in number, 1 The limited CI calculations carried out employ a core of six MO’s (doubly occupied in all configurations) in each case, three of which (lal, 2al and 3al) are common to all treatments. The valence set (MO’s subject to variable occupation) consists of from seven to nine MO's, chosen in such a way as to best afford improvement in the representation of the parent state relative to that in its SCF wavefunction. The specific choices of core and valence orbitals in each of the six CI calculations undertaken are given in table 3; the level of CT.employed for the H2CG states in this work is thus roughly the same as that used in the aforementioned ethylene calculation [Z]. Technical details of the CI procediire employed are reported elsewhere [8]. All single and double excitations relative to the respective parent configuration ‘m each case are taken into account; in addition selected triple and quadruple excitation species are considered, these being selected on the basis of diagonal energy. The orders of the resulting secular equations solved in each case are also given in table 3; in each calculation only singlets and triplets of the spatial symmetry of a given parent configuration are given explicit consideration In general the CI treatments considered in this work are more effective than those used in the previous H2CO calculation, as can be judged by the greater CI energy lowering obtained for the ground state (0.0564 hartree or 1.53 eV, compared to 0.0327 hartree or 0.89 eV). The differences in energies between the ground and various excited states obtained in these calculations are given in table 4; the method of selecting the appropriate total energy for a state obtained in more than one CI(PCM0) treatment is the same as that discussed in I, the principle being simply to select the lowest set of transition energies corresponding to a mutually orthogonal set of wavefunctions. It is not immediately apparent that the IAI - % transition energies selected in table 4 fit this description, since they result from different CI(PCM0) treatments, thereby raising the question of whether these states are all mutually orthogonal; similar reservations would_seem to be called for in the case of the 3Al - X species in the table. Inspection of table 3, however, shows that there is very little practical difficulty involved in this point since the va.Ience sets employed in the two pertinent calcul&ions (l!Ios_ 1 and 7) are very

,

.,’

‘_

..,:

‘.

. 1

131

+&tine

8‘. n,uinder 1

CHEMIC-AL PHYSICS LETTERS

: : n&ily sutu+lly’ ex&ive $. ..-, The present CI(PCM0) transition energies .;are.‘aJi substantially higher-than tine corresponding -,, SCF values, as is to be expected; they are

i,h’ e overlap integr’al between the ground state CI‘wavefunction (resuIting from talc. No. 1 id that of the n.- 3pn IAl Rjdberg state is iesd than 0.001; a Gnilar value is ,obtained for the overlap between the -7i 1 R*-IAI Cl function -and the n ‘3pn species. and of course,. the ground state and ST-+s* singlet are 1 rigorously orthogonal since they do result from the ..aame.CI treatment. Detailed analysis of the two -3A ‘CI w&efunctions referred-to in table 4 shows thai.these species are also very nearly orthogonal;

-, . < Constitution of tht: MC tiasis sets for the series

Table 3 of Cl(PCM0) calculations

for the ground and excited states of H2COa)

2

4

5

7

8

C

C

C

C

C

V

C

C

C

C

c

V

1

Calc;No.

.l danutiy 1971 _a&o soma&ai l&her t&i the previous CI results obtained in I, reflecting ,the. eff ect of the -_: more extensive ground state CI treatment. The -present data are seen to overestimate the experimental transition energies [?I (where comparison is possible) but the discrepanciesare uniformally in the order of only about 0.2 - 0.3 e’f; these comparisons between experiment and theory are summarized in fig. 1. PosJltive identification beJween “a and 3A2 (n - x*), A and lA2 (n - 8*); B and lB2 (n 2 3sC), C and lA1 (n - 3pnC), and 6 and ‘B2 (n - 3pub) is indicated in fig. 1;clearly the CI(PCM0) transition &lergies are in much better agreement-than their SCF counterparts. The experimental band ,

.’

MO‘s

V V

Bal.

V

.‘-.9al

V

.lOal

V

llal

,

.I% lb2 2b2

C

V

C

V

V

v

v

V

v

V

V

3b2

4b2

v

V

.V

“b2 -, 6bZ.

”

v

V

V.

V

v

v .v v-

.-

:

Sbl

‘, __

6bl

-...

: ..

:

_ . ....

v-,.

C

V

v

I

V

v

V

.v

v

-_

._v

. . . 1,76(1A,) - ‘206(3Alj

V

V

V

‘v

i+epA;)

.: 21@‘(3Ai) .:.

‘c.

C V

V

V v ,. 2

c‘

v

.,

‘17+A1) 266&S;)

.

V

.

-v

.-.:,,

:

--’

_, .,170(1B2)‘

,-

’

210(3B2):

..

.’

__ ‘-:- _l.?6(1A,)

._.’ _165(3+) -.. __. : :

. ..

V

: !68(lBl). :L_2f2(3B1)’

].,

;,-.

_..

._--_

-

1 January 1971

CHEMICAL PHYSICS LETTERS.

Volume 8. number 1

‘Transition energies (eV) of

Table 4 obtaLnedfrom several calculations cornpared with experiment

H2CO

SCF

d1 (PCMO)

CI (PCMO) from I

--

Expti. ref. [91

WH ref. [61

State

Description

CaIc.tio.

lAl

Ground state

1

0.0

0.0

0.0

“x0.0

n-n n-b

2

2.25

3.41

3.38

3 3.12(3An) (3.12-3.44)

2

2.62

3.61

3.80

x 3.50f1A”) (3.50-5.39)

5.56

5.66

3A2 lA2 3Al

8-a

3B2 lB2

**

t

.1

rid 3s C n--3sC

5.66

(5.69) a)

4

4.L7

5

6.03

7.32

5

6.07

7.38

7-48

(n - 3sO) 8.09

8.10

8.11

8.30

2 3Al

n -+ 3pnC(b2)

7

2 ‘Al

n - 3pnC(b2)

7

3B1

U-r*

8

6.96

8.14

7.62

lB1

o-+?r

8

7.86

9.03

8.61

6.93

0 - 3pq$

*

2 3A2

n + 3p*C ($1

2

9.06

2 ‘A2

n - 3pQ(bl)

2

9.07

Z3B2

nd3pU C n - 3poC

5

8.29

5

8.39

1

11.41

2 lB2 (3jlAl

11.72

g ?.i)8(LAn. first Rydberg) 7.08-7.5L, rc=3.n --nsal) c 7.97(LAl. 2nd Rydberg) (rt=3. n - W+z)

9.35

11.31

(‘B 3rd Rydberg) 5 8114(n= 25: n - npal) -

a) Alternative higher value for the 3Al transition energy resulting from the CI cakuLation based on the SCF MO’s of its parent configuration. j?;, 6 and 5 have no ca.lcuLzted their description. requires the use of 3d, 4s and 4p functions, not incIuded in the present basis. The present calcukted transition er,ergies are also in good agreement with the results of Whitten and Hatieyer [S] (table 4). It is cert&nly interesting that this agreement obtains even for the Rydberg states, for which the present calculations lead to a description of their upper orbit&s as predominantly carbon, systems g,

analog

since

while th~~cakulatious of the Iatter authors, which do not use ali the appropriate carbon basis functions, identify them as mainly oxygen in character. The inctication.from tbi8 result is that in the description of diffuse MO’s the nature of the A0 basis‘ is not nearLy SO critical as it is for the representation of valence species. From

. ;z .-

_&q?-;, 3A*(“-d

:

i.0.: ‘_ .-

,_’

. .

.

. __-

:E:rpt . . . -‘I.,,

a qualitative point bf VieW, since the radii of the diffuse orbital are so much greater than the C.0 tj&id.d&km~e in formaldehyde, there is a .real qWStiOll a-t0 tb& meanfr@ulze& .of refer.I_

,’

,. -: :, ‘S‘- ,. ,__,

_.. Cl(_)

.

‘Y

,.

.-

.

’

.,.:

.

.

,--,“.’

-

Fig. 1. C.om&rigon kf,&LcuLated~andexperimental -. ‘,, Xrtisitioo .eneTgies of H2CO.

-:-.

,- _ : -- .:_ ...: .. -: . (,‘_ _,:-’ ;., ,‘_‘I __ : ‘3:

._

,I

_,-

.’

-.

133

;~,~‘:-‘-~&;ini~~~-~&n&jr ,l,).

-. q

.::

. .._. .. . .. :

rmg

::.,:.

--

Ji;:;::. . .._

,_,

‘, . ,:

:,

to”these- ori@als

. . .-

&l

.‘.

-. : ._‘. ::.: -^. _, : :_:..:

_.:

-~.

._

.: ._

:._

&bon.

:‘I’: Oxygen qiiecies;.:Iii.any.‘e~ent .t$L &&k

.,,.. ~:so@. of the tRio:types’ of.,sp&tr&

_.

_,

.I-.

L

‘_,

.‘.

‘:;__

.jCHEEnICA_~~,PKPSICS-LETTERS,. _~_ :‘:,:,.:‘:.;;,;’

._

b:einp&zsr

,.

:..

__

-,

...

-

&;&;e& &: : _I:‘.:,

-.

,,

..--

_

;

..,

,’

,,.‘,‘.

.-‘,I,.

,.

-..:-

1

”

.__

:

‘. -, /‘. $&y~~e_u~$&r-statr1: .

ijiotiIbljr &pai&

‘_.

: : ~~_,J+&$~~~~

._ .: : ;‘:’ :

&rnpa>L~ . .” “pte&ial-&jr+&’ iS _+z&s&f& :aS &xl&& the i&cu-~~

c&uh&ons : 1.. -- : lations of 3 h&e found. .~E&it&tir;s’~ to &e .pair of _See+rio’&z$port t+,hypothesis disrmssd]inI,‘:‘.,., : ‘. Rydberg 3~,~~2~-Stites &h&ted ‘at 9.9 eV are. ‘. : :namely thit a.se?ies.of. rel$ively-qmall ? .. ,. :I--. -:~,‘:f&bidden acc&&g.tq _bllh’dibole se@tioxx~ru&; ; : CI(PCM4) treatm+tsl for certciin important 1 ‘! .] _- a f&t_ which: und~~bt~d~y~-~~~s’ thttir absence .. .,.. ‘parent. c&&&rations’ &I be just ti accurate.& 8.: .. in the H2CO- ~pe,otr&n:$:~ .. ‘. .I. : j’ :. 1Fin+&; the difference betweenthe; &ounb ’ ,- much la~ger;_~T“~ed~rnent.‘~~~~oyini:,a trial- I. : ..-.~d~e~ror,co~~~ratio~ sele_ction process., i’ -. -:. 3&&d CI &n$?rgy and the 2Bi $CF value~o&bl& --.-, 2 for, the positive ion.+ll.OCl ‘eV, in good.agsee-, : Qie of th.emost. interesting results of table 4’ .:. .‘) .,is.the f&tit that the upper state in,.the.lokest :ment [email protected]~,th~‘minimum 1.P. of H2CO; (10; 8s ev), ‘: .,e,nergy ‘.;Pi-.ji trai+ition !.5 not;. as origin&Hy.- ,which is the limit of the various.,Rydberg series ..thought [IF], the $I- &*,s$,eci$s. $&toad.the related to single excitations from thc’n orbital. ‘. : As in the .&se .of. ethylene [ 21; ‘.a. CI treatment‘ for : .’ .$t+ng absorption at 810 eV ‘(X - C). is’now’. .: beHeved to‘ correspond, to -an n - 3pb2 transition : the positive ion which would be’ equivalent to .’ -f9]; the transit+. to the-n::? r.” singiet-apparently. _’ that car&d out for the ground state~wouid be . Lies much: higher, .in the 11-12 .eV region of the rdatfvely ineffective’because’ of Brioullin~ s t@eoreti; in general it ‘is not’ Obviou& what level electronic spe&um._ The apeement between experimentand theory in thisease prompts one .of CL!1 for the ion constitutes a&equivalent treat: to ,cax~i-yavex. this- rehult td .the,study of t.iik. :- tient, to that emplbyed for- the neutral molecule ‘, ethjrleno’spectrum; it seems reasonable that the sipce &I this case there is in eff&t ilo upper ‘_‘. orbital. a+ ~~‘~aZenc~‘-singlet-shouId occor at +n ener,gy .’ ..:._’ approx%m&ly’d. 5 - 1.0 eV Lower than that of its. .’ ’ ’ H2CQ analog,~ and caiculatidns for ethylene [2] ‘. ‘,-in fact, obi;8in this:r+ult, (c+ulat&d transition 4. ~ONC~~~ON _’ . .. :_ ::_ene&& is 19. ?:eV),- The .an&og to the n - .3pb2 I :- tr$tisitiosi bf formaldehyde might then be The CL(PCM0) trqatment,’ as in previous ._ chosen as. the lblg --G2bzu Rydberg species; b;tt pclork with ethyiene, obtains good agreement fith ‘, ez+er,ixq@I transition energies; discrep&icies : s&-icein -etbytene the’ highest occupied ground of’+O.2 - 0.3 &V are noted fbr each of the first .’ st#e.MO is-the B orbital,. a tiqkg logical choice ’ is. .another S’-.i;* species of ethylene in .which .: ‘. ..’ five transitions (two valence and three.Rydberg 1‘the .T*- is-r&tively-diffuse. ‘Calcu~tions find a species). The fact that tho’caiculations overestix’+ ~*.‘v&ticai excitation in the same general, ..., .: mate transition energies in the present C~(~CM~) t+@ment. but slightly underestimate them in a :&gicjn ‘of the’spectrum -(8.3 eV) but the: n! MO .. ; :- in %I& case is found t0:b.e.a: hybrid. df ~vtience. previous series.of calcUlations emphasizes that ; r+d Ry&&g &a&t&--aHko, aS indeed otio .the choice of the $pedifii: .CI treatment undermust: expect because of the inevitable &king “. I taken can affeck-the qu’&titative results, but the ‘-. : mdication * ‘betuie&tho two n.* MO’s of the same symmetry. 6 both the ethylene and form.aldehyde ..A

Volume

8, number 1

-.

CHEMICAL

PHYSICS

CI procedure whkh.m&tains a relatively large but equivalent core for all the interesting states of a given system should sue- : teed iu balancing the correlation error in the SC% procedure. Employing the SCF MO’s of each parent configuration for its own Cl expansion and those of closely related higher energy species thus allows one to achieve the goal of calculating reliable transition energies ( and excited state wavefunctions) without the necessity of using large-scale CI. techniques, in which complicated procedures for seldcting configurations must be followed and/or relatively-highorder secular equations must be solved. Hence.=

ACKNOWLEDGMENTS

The authors wish to thank the Deutsche. Forschungsgemeinsch

for financial

support;

the services and computer time made available by the University of Mainz and the University of

i

LETTERS

Nebraska

1 Jacuary Computer

denters

L971

are gratefuLLy

acknowledged. REFERENC& [l] R, J:Buenker and S. D. Peyerimhoff. J _Chem. Phys. 53 (1970) 1368. [2] R. J. Buenker. S. D. Peyerimhoff and 1%‘.E. k’zrnmer. J. Chem. Phys. . to be puhlished. (31 J. D.Petke. J. L.Whitten and A.W.Douglas. J. Chem. Phys. 51 (1969) 256. .(4] S.Shih. R.J. Buenker. S. D. Peyerimhoff and B.Wirsam. Theoret. Chim. A&a (Berlin} 18 (1970) 277. [5] T-H-Dunning Jr.. W.J.Hunt and W.A.Gcddard III. Chem. Phys. Letters 4 (1969) 147. [S] J.L.Whitten and hI.Hackmeyer. J. Chem. Phys. 51 (19691 5584. 1’71C.C.J.Roothaan. Rev. hlod. Phys. 32 (1960) 179. [Sj R.J. Buenker and S. D. Peycrimhoff. Theoret. Chim. Acta (Berlin) 12 (1968) 163. [9] G. Hersberg, hIolecular spectra and molecular structure III. Electronic spectra and electronic structure of polyatomic motecules (Van Nostrand. Princeton. 1966). [lo] J.A. Popie and J.W.Sidman. J. Chem. Phys. 27 (1957) 1270;