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The spatial partitioning and transferability of molecular energies
CHEMICALPHYSICSLETTEa.
Volume 8, number 1
THE
SPATIAL
PARTITIONING
AND
MOLECULAR
1 January
TRANSFERABILITY
19il
OF
ENERGIES
R. F. W. BADER and P. M. BEDDALL Department
of Chemistry,
McMaster Received
University,
4 November
Hamilton,
Onfauio.
Canada
1970
This note examines the question of the transferabiiity of the energy of a hnded different a spatial
fragment between _._ molecules in terms of the transferability of the charge distribution. This is accomplished through partitioning of both the charge distribution and the energy of a molecular system;
1. PARTITIONING OF THE CHARGE DISTRIBUTIONS 9m
..a
9.m
We have recently proposed a partitioning and a classification scheme for molecular charge distributions, each scheme being directly related to observed (i.e., calculable) properties of the charge distribution [l]. The form of the charge distribution suggests a “natural” partitioning in the form of a plane P, perpendicular to the bond axis through the point on this axis at which p(r) attains its minimum value between a pair of adjacent nuclei. This procedure is illustrated in fig. la for the CO2 system. Integration of p(t) from infinity on the left up to the plane Pm defines the total electronic charge population on the left-hand oxygen nucleils, to. Integration of p(r) between the planes Pm and Pt; defines the total electronic charge population on the carbon nucleus tc . In addition, a further plane PA through a given nucleus A partitions a total population into bonded @A) and nonbonded (nA) contributions for a terminal nucleus, or into a number of bonded populations for a multiply onde centre. 8 8 ThusinCQ,tO=bO+ngandtc =bC+bC’ One of the‘ reasons for defining the above populations is that changes in a system caused by a change in the bonding environment are largely restricted to either the bonded or nonbonded regions. ThFs the principal changes in the charge Total molecular charge distribution for dO2: (a) constructed from a near H-F wave function [3] for COz, (b) constructed from the charge distribtitions of two $0 fragments using a wave function of comparable quality (3). The numbers denote the total populations (to and tC) as calculated in (a) and as predicted by:the fragment populations in (h) (see ?ahle.‘la). ,’ Fig. 1.
..
.-.
-:
_. ‘. : -_
I _
.
29
. ..
Volume-S.
number
-_ ..._ -_i--
1
CHEWCAL
----
PHYSICY
Pop&on
---._ ----
LETTERS
~Z&L3ons
1 ~Januarjr 1971
a)
.---Total.bonded populations
Bonded and nonbonded contributions A.
B.
ABC oco OW
“A 4.24 4.16
bA 4.76 4.76
b* B 2.00 2.03
bG %
bc
%
W%)
2.00 2.03
4.76 4.76
4.24 4.16
6.75 6.79
6.75 6.79
FCN
4.64
F$N
4.65
5.07 5.14
1.95 2.02
2.33 2.35
4.30 4.25
3.72 3.69
7.02 7.16
6.63 6.60
HCN H@N
0.43 0.51
0.62 0.66
2.65 2.60
2.31 2.35
4.28 4.25
3.71 3.69
3.27 3.26
6.59 6.60
WNO NqrO
3.65 3.57
2.96 3.43
3.91 3.43
3.lG 2.90
4.05 4.34
4.27 4.16
6.87 6.86
7.21 7.24
b(CC’)
h(C’H)
nA
bA
bc C’
6;s
0.45
0.63
bA C 2.76
bc’ C
HCCH b)
3.17
3.17
LiCCH
1.05
1.16
3.49
2.69
3.68
- ACC’H
nH 0.45
2.79
0.66
0.48
6.34
3.39
6.37
3.45
4.66
5.09
2.01
3.73
2.64
2.79
0.63
0.44
6.37
3.42
ClCCH
8.54
8.39
2.84
3.50
2.85
2.76
0.64
0.44
6.35
3.40
ABC
“A
b(AW
MBC)
4.23 8 .Ol
bA B 2.01 3.80
bc B
oco SC0
bA 4.76 7.17
FCCH
C.
2.76
bH 0.63
W’C)
b)
2.01 2.03
bC 4.76 4.77
6.77 10.97
a) All polyatomic wavefunctions in A and B are from ref.[3] and all are of near Hartree-Fock accuracy CICCH, The wavefunctions for CICCH and in particular for CO2 and SC0 of part C are of comparable (doubIe-zeta + polarizing functions). The diatomic functions are from P. E. Cade (see ref. [l]). b) Compare F-4 and H-q values with corresponding fragments in A.
distribution in terms of populations may be spatially isolated, and other populations undergo only minimal changes. One example of this is illustrated in fig. lb which shows a charge distribution for CO2 obtained by discarding the nonbonded population on carbon in two CO diatomic molecules in their XIC’ ground states and joining the two fragments at the resulting planar surfaces through the carbon nucleus. The populations, (see table la) both the totals and the bonded and nonbonded contributions, differ at the most by O.O8e-, between the actual molecular charge distribution (fig. la) and that constructed from the fragments (fig. lb); the total bonded pop lations between carbon and oxygen b(C0) = = b $! + bg differing by 0.04e’. The total population on-C in the diatomic species CO is very asymmetrically.partitioned between the bonded and nonbonded regions with nC, = 3.05 and bg : = 2.03. In addition, the nonbonded charge is in the_fckm_of a very diffuse distrection of relatively large spatial extent.-_In spit$of- the very different nature of the bonded ax@ nonbonded PO-. pulatiks, the .change in the populations bet$een CO and CO2 is almost entirely restricted to-the . .
6.77 6.50 except for accuracy
nonbonded fragment on carbon which in this case is set equal to zero (i.e. discarded). The small differences in the populations between CC2 and the CO fragments are surprising in the light of simple valency concepts which ascribe a triple bond and single pairs of unshared electrons to each nucleus in CO and double bonds and two unshared pairs to each oxygen in CO2. There are no changes in any of the populations which would correspond to such a gross transfer of charge from the bonded region of the diatomic CO to the nonbonded region of oxygen in the CO2 system. In table 9a a number of examples are given comparing the populations of ABC with the fragments A$3 and $C!, the vertical bar designating the fragment being transferred. In CQ, .FCN and HCN the total populations and the bbnded and
nonbonded contributions show little variation be tween the diatomic fragments and the polyatomic system. In NIT0 the totd populations do differ significantly between the two cases. .However, the total h&d&d populations b(q = bN, +bNY and &(Nq) = bN + 68 and the nonbonde %, popu. “I ations are little changed. Thus_ wh.en there-is a considerable‘ change @I the populations due to a’ ._ . . ..-_ .’ ., ..; _:. ._. .’
Volume 8, number 1
CHEMICAL PHYSICSLE’I’TERS
change in bonding, the change appears t-o be restricted to adjacent bonded populations. Thus charge is shifted from 6& - q and from bg - bg compared to the diatomic fragments, but the nonbonded and total bonded populations remain nearly unchanged. This shift of charge density wiflrin a. bonded region is further illustrated in table 1d:in which comparisons
are made between the populations
of a number of substituted acetylenes ACCH and the parent molecule. In these exam les the F +2 bonded population b (CC’) = bC’ differs by only O.O3e- over all four mo ecule!?‘but there is considerable change in the separate values of bc’ andbC from the symmetrical case of HCCE. ‘The dag in tables la and b indicate that the nonbonded and total bonded pop&atiozs are rather insensitive
to changes in bonding and may be
poor gauges of changes in properties. In none of the above systems is there any question of transferability of energy for example, in spite of the near constancy of the individual populations such as is found in CO and CO2. Close examination of figs. la and b shows that the charge distribution irt the bonded region of the ($0 fragment of the CO diatomic (fig. lb) is more contracted along the bond axis than it is in CO2. Thus the principal reason for the bond energy of CO being much greater than the average CO bond energy of CO2 is that the charge density is more contracted in the bonded region of the former compared to the latter. Charge populaticns (as defined here) undergo relatively small changes as the bonding environment is changed. The changes in properties
mining the properties of such Fragments. Thus we propose that the extent to which the AB rend
BC bonds of A2B and C2B are transferable to ABC is determined by the extent to which the charge distributions of the fragments A8 and I@ are identical, up to the plane PB in each fragment, to that found in ABC. The energy of OCS may be obtained additively from the energies of CO2 and CSZ to within a Eew kcal/mole [2],
E(OCS) = Q[E(CO2) +E(CS2)]
& 2 kcal/moIe
.
The populations of the CO fragments in OCO and and OCS are given table lc [3]. The changes in the bonded and nonbonded populations of the 60 fragment are small tid the total fragment popuLations differ by only O.O134e-, being largest in OCS. Fig. 2 compares the charge distributions of the $0 fragments in the two systems. in this case (unlike the case of the ($0 fragment in CO2 and CO, fig. 1) the differences in the distribution of charge as well as the populations are smalL.
caused by a change in the bonding
environment aye caused by a relatively smal2 redistribution of charge density within a given spatial region and not by a large cizmge in its population. A simple example of transferability of bond properties (bond additivity) is transferttiility of the AB and BC bonds in certain symmetrical systems ABA and CBC TVthe unsymmet:rical system ABC. The above results indicate that the planes through the nuclei represent at least partial barriers to the transfer of charge. A perturbation may be transmitted through the barrier causing a redistribution of charge within a bonded region (eg. $0 : TO) but very little change occurs in the total populations a.8 defined by these planes. These results suggest that transferability, when it does occur, should be reflected ln the iear or total constancy of the distri&&.ion of charge (not just the populations) in the fragments A$ and $C and that on65 could obtain a set of transfe*le properties b:y deter-
Fig. 2. Comparison of the ctige fragment
distribution of the 40 in SC0 (solid lines) with that in CO2 (dashed
lines}.
-31
Volume-8, number 1
CHEMICALPHYSICS.LZTTERS
It is impqrtant to note that the principal difference in p(r) for’ $0 b&ween CO2 and OCS occurs at t.& boundary plane in the outer diffuse regions of the Since p(r) must be single charge distribution. valued akd continuous at the boundary plane, such small changes in p(r) at the boundary will occur in general. They would be absent only if the radial distribution of the charge density on carbon in the. boundary plane was identical in both the a0 and $S fragments in their parent molecules. This situation is not obtained here (p(r) is more diffuse in $S than in $0 at the boundary) and is not expected in general. The question to be asked is whe!her near ‘ additivity of the energy is a result of a near transferability of the charge distributions ‘ of the fragments as defined by the plane PC; i.e., is theplzne PC the correct boundary surface for .the explanation of additivity ? To answer these questions& is necessary to define and determine the energy of a fragment Al& This is the subject of the following section. 2. PARTITIONING
OF THE ENERGY
To effect a spatial partitioning of the energy it is necessary to express the &ergy of the system as.a function of the three spatial coordinates defining the charge distribution, i.e. one must define an ,energy density. It is possible to write such an expression for the kinetic energy of a system and we have previously given definitions for two kinetic energy density distributions [4]. If the charge distribution is expressed in terms of the natural orbitals of the system: p(r)
= Cii$(r)
eibi(f) =Cf+)
&,d:
=Ur) -. z(A .‘
+ G(r)
,
(1)
.’ = - v+!rj
theorem to both sides of the
-.
form of eq. (1) with G(r) transposed
JK(r)dr-
JG(r)dr=JL(r)dr
(4)
,
we obtain
(5)
where d$/dz and dp/dn are the normal derivatives (outwardly directed) at the point dS on the surface S. Eqs. (5) and (6) are of course equal, the form& being written to illustrate that it applies separate11 to each natural orbital in the expression for p(r). If the boundary surface S is taken at infinity where both 9 and pand their normal derivatives are
Ai
,
January 1971
we obtain
the results
=0 andJK(r)dr
[4]
=IG(r)dr=F
.(7)
The
where ?-is the total kinetic energy of the system. However,. it was also demonstrated thatK(i) m2y be expressed in terms of the laplacian of p( r) and tbe._gradient+ of the orbital densities Pi(r) as [4]. .__
integrated
jL(r)dr
-' 'fK(r),dr = F ,
,+
Applying Green’s
zero
-then the mosi immediate definition of a kinetic energy densiiy is given by : K(r) = -+C Ai$i(r)V.2@i(r) , i for which one clearly has
:whtire
1
6) .
L(r)dr
= $$~~~podrrdr/~=~~~
,
(8)
where p(r) is expressed in cylindrical coordinates with .z lying alocg the internuclear aXis in the direction of negative 12&rough the plan?r boundary surface .and p!‘(z) is a one-dim‘ensionall representation of thetotal. charge. distribution, i.e., -p’(z) dz is-the total charge Iying betweek the two parallel planes at z and z + di Eq. (8) restits since dp[dn of eq. (6) c&be finite only in,thti perpendicular bouncltiy p&ie it nliciegs .B;: Th& the value .of .-
Volume 8. nu,nber 1
CHEMICALPHYSICSLETTERS
slope of the one-dimensional representation of the charge distribution p’(z) at the position of the boundary surface and its value for the adjoining fragment sharing the same boundary surface must be equal and opposite. Recognizing that Green’s theorem is a special case of Gauss’ theorem we are Led to an interpretation of the above equations through an identification of p(r), the charge distribution, as a scalar field. Thus -4L(r)
= V20(r)
is Poisson’s
= -4[K(r)
- G(r)]
I January 1971
However, the same conditions hold in another situation, one of direct consequence to the transferability question. Consider the transferability of the fragments A$ and BC from ABA and CBC to form ABC. Clearly, since in the symmetrical parent molecules A2B and C2B dp’(z)jdz = 0 at the position of the central nucleus, the integral of L(r) dr over the regions defining the fragments A$ and I$C will be zero. Thus in these cases we have &K(r)
dr = &
G(r) dr = T(AB) = $ i
;
equation and its solution yields
(11) s L(r)dr
=0
,
A@ that is, one will obtain one-half of the total kinetic energy. Furthermore, the virial theorem will apply inasmuch as where A = 1 r - r’/. Thus the total charge density at any point in space may be considered to be the scalar field arising from the source -4[K(tI - G(r)]. Application of Gauss’ theorem
to eq. (9) yields eqs. (41, (5) and (6) directly and
enables one to interpret dp/d?z as the vector field or gradient af the scalar field p(r), and the surface integrals involving dp/dn as yielding the net outflow of this vector field over the surface S. In general because of eq. (8), integration of G(r) dr or K(r) d r over some restricted volume of space will yield different values for the corresponding kinetic energy contribution for that region. Thus the question arises as to which function should be integrated to determine the kinetic energy of a given spatial region. Furthermore, we wish to make the assumption that for a system at internal equilibrium such that the virial theorem applies, we may equate the negative of a kinetic energy contribution from a given region of space, -FR, to ER, the contribution to the total energy from the same region. Let us consider both these questions, the choice of G(r) or K(r) for determining TR over a partial region of space, and the-equating of :FR with ER, in terms of the transferability problem. There are two situations in which we may answer both questions withcertainty. The first is the trivial one obtained when-the region of integration R is set equal to the whole of space. Inthiscase .-
$T = -$E = -E(A$)
.
(12)
The total energy of ABC will then be given by E(ABC) = E(AB) + E(@C)
.
For absolrcte tmnsfe?-Gbility to obtain, the
charge distribution of each fragment must remain unchanged in the construction oE ABC from ASB and C8B. Thus the integral of L(r) dr over the fragments A@ and $C in ABC must stiIL equal zero. (It is worthwhile here to recall tie theorem of Hohenberg and Kohn which states that for every charge distribution there is a unique value for the energy [5].) However, absolute transferability as defined by the conditions in eq. (11) will generally not be obtained as can be seen immediately from the property of the integral of L(r) dr as given by eq. (8). In ABC the integral of L(r) dr over either fragment-will deviate from zero as p’(zj will, in general, no longer b.2 a maximum at the position of the boundary plans. As discussed earlier in terms of the examples of CO and CS, even in cases where additivity is anticipated small changes in p(r) at the boundary pIa.ue are to be expected, as otherwise one is demanding an identical radial distribution of charge in the boundary plane for nucleus B in all of the systems of interest. Thus in general jk(r)dr
R
Cj- G(r)dr
R
for either fragment in ABC, the charge distribu-tion and the energy of each fragment is altered
and absolute transferability is not obtained. What we wish to demonsh%te is Mat even in tie :
:
-
._.
::
33 ‘_
: Volume 8, number 1
CtiEMXCALPHYSICS
absence of absolute transferability of p(t) an a.Imost exact additivity of the energy can still be obtained if. the smaII unavoidable changes in ~&,rf assume a special form. Additivity of the energy will be obtained if the values of either IB K(r) dr m &G(r) dr over both fragments remain unchanged from their vaiues in.the symmetrical parent species, since .:’ .$ &)dr
i- &K(r)dz’
A@
=
. _j-$(r)dr
+ &-G(r)dr
=
-E(AjW .
.If we make the assumption that &K(r) dr over a fragment may be identified with the kinetic enerG of the fragment, then the extent of additivity will ,be determined by the extent to which the change in Jd&(r) dr caused by the transfer
is absorbed
by a change in J”K(
r)dr rather than in &G(r) dr for both frag-
ments. The physical justification of this particuIar possibility as being the explanation of near additivity of the energy is as foliows. Parr and Brown [S] and later ‘Nelander f?] have shown that the virial theorem for a poiyatomic molecule may be written as
LETTERS
.1
Xn summary, for near additivity the small but necessary changes in p(r) caused by the transfer result in small changes in the forces acting on the nuclei and hence in the equilibrium bond lengths. The extent to which these changes, as measured by the deviation of [do’(z)/& 1from zero, are absorbed as changes in the kinetic energy and hence inlRK(f).ds fragment,
and
notdin &G(r)
< pAB(ABA).
Since
s K(r)dr A@
= FA$,
both this
I,( P) dr must decrease from’their Integral and s A$ vafue in ABA on transfer to ABC, and in particuIar $ L(r) dt < 0 in ABC. Since $ L(r) dr A@ A$ = $&) dr , it follows that JBCK(f) dr (=T;tac) , -: must incr&ke over its value .in CBC and hence
RBc(ABC) 84’ ..
< RBc(CBC)*. ‘.
’
dr for each
determines
the extent of additivityof the energy. Xg other words, the proposal is one which accounts for a nearly constant total energy or total kinetic energy on transfer (the sum of the integrated G(r) dr values} in spite of the small but necessary changes in the kinetic energy of each fragment separately. Other possibilities, such as changes in p(r) but not in &,&C(r) dr for each fragment would appear to be in violation of the theorem of Bohenberg and Xohn [5]. Substantial changes in the Integrated values of both K(r) dr and G(r) dr for each fragment (reflecting substantial changes In p(r) on transfer) which happen to cancel out on addition in ABC is unIikeIy and could be classed as an “accidental” additivity. We now give a numerical example illustrating the above proposals using the wave functions of McLean and Yoshimine [3] for the transfer of the $0 iragment between CO2 and SCO. The charge distribution
of $0 in CO2 yieIds
f K(r) dt = $$ G(r) $0 where the sum may be taken over all the bonds of the system [7]. The small adjustments in p(r) required at the boundary plane will result in an ,externsI virial contribution to the total kinetic energy of ABC from each bond. Thus in general RAB(ABA) f RAB(ABC) asRAB must change to a new equilibrium value in ABC. Consider the case RAB(ABC) > RAB(ABA). Then TAB(ABC)
January 1971
dr
= 93.7138 au
and $6 t(r)dr
= 0,
i.e., dp’(z)/dz = 0 at the boundary plane. In SC0 (as suggested by fig. 2) p’(z) slopes off towards the oxygen nucleus at the boundary plane, dp’(z)/dz is negative and hence = -0.0343
au or -21.5 kcal/moIe.
Ir’o L(i)dr We then correctly surmise that. R&SCO~ (=2.2016 a$? RcOfCO2) (= 2.1944 au) and that Z’@SCO) c Z@-J(CO~) as determined by . * This argument predicts that ia systems where one obtains near additivity of the energy, one bond length . must increase, the other mtist necessarily decrease on transfer of the fragments to ABC. This is SO for qnmple,
in the COgI CS&SCO
1’.
‘.
system. -.
Volume 8, number the integrals
CHEMICALPHYSICS LETTERS
1
of K(r) dr.
The value of
1 K(t)
dr
does indeed decrease in value on transfer (to 93.6774 au) and its decrease is equal to the decrease in s L(r) dr to within 1.5 kcal/‘mole. CO Thus the changes in the charge distribution of ($0 cn transfer as illustrated in fig. 2 are of such a nature as to decrease Ike kinelic mcqy of the system whiIe Ieaving the value of JGoG(r) dr almost unchanged, its value in SC0 being 93.7116 au, corresponding to a decrease of 1.5 kcal/mole in the energy of $0 on transfer. The value oE 1 L(r) dr in SC0 is + 0.0343 au SC indicating an increase in the kinetic energy of the Se fragment on transfer from CS2 to SCO. If this change is absorbed almost entirely by the change in the kinetic energy of the $C fragment on transfer (as found for the $0 fragment) one predicts an additivity of the total energy to within 2 or 3 kcal/mole in the COq,CS2-SC0 system. The standard (double) bond energy of the $0 fragment can be defined as E($O)
= E(0) + &F(C) + s G(r) dr
$0
where E(0) and E(C) are the energies of the ground state atoms with the integral evaluated for CO3. Other fragment properties could be defined, the dipole moment for example. Allen and Shull [8] were the first to point out that the electronic and nuclear repulsion energies of a system are not separately additive and that only through the use of the virial theorem, which relates the electronic kinetic energy to the total energy of the system can one provide a theoretical justification for additivity. Allen and Shull presented their arguments in terms of the transferability of the geminals defining an electron-pair wave function from one system to another with the assumption that the geminals could be so chosen as to be localized in the regions of the individual bonds. A wave function suitable from this point of view is the separated electron pair function proposed by Parks and Parr [9]. Lykos and Parr [‘i’], in discussing-the u-x separability problem have suggested that the total wave function may be expressed as a partially antisymmetrized product of two functions, these two functions themselves being antisymmetrized functions, one for the u electrons the other for 8. electrons. One can consider applyingthis concept of separability to the molecuIar
I Janun~ 1971
fragments as defined here by the boundary plane through a nucleus. Thus *(ABC)
= Jj,!/(A~>~($C)]
.
where p(A$) and $@C) are to be transferable antisymmetrized futctions for the A$ and $C fragments and the brackets denote a further partial antisymmetrization corresponding to the exchange of electrons between the two fragments. The present analysis. however; the transfer of such fragment wave
suggests
that
functiclns
between the different systems (all nt irttental eq:iilibrium) will in general not be possible. In order that the charge density and the wave function be single-valued and continuous at the boundary between fragments, changes in the wave function are necessary. (These changes may lead to a non-integral value for the electron population of a fragment.) Thus a necessary (but not sufficient) condition for absolute transferability of a fragment wave function between different systems is that the flux of Wp(r) for the fragment through the partitioning surface as defined by eqs. (5) or (6), be the same in all of the systems. En the case of linear systems this requirement is simply stated (and simple to test Ear) that dp’(z);‘dz at the boundary surface remain constant. As pointed out previously this is a stringent condition and one unlikely to be met in practice. Rather, what we have proposed from our considerations of the behaviour and properties of the charge distribution is a mechanism which acccunts foF the additivity of the energy between systems in spite of the small change in p(r) of each fragment which occurs on its transfer between systems. If the changes in the bonding environment of a fragment are remote, then the corresponding changes in its charge distribution on transfer couLd be vanishingly small and such favourable cases could approach absolute transfeiability. One could consider the possibility of preparing a fragment wave function for transfer by first placing the parent system in a non-equilibrium configuration. For example, the transfer of the shortened $0 fragment from a sLightLy asymmetrically distorted Co2 (to obtain a dp’(z)/dz < 0) to SCO. This and other questions raised by the present approach to transferability, together with more examples are to be reported. In addition, the usefulness of a spatial partitioning of the total kinetic energy in predicting and understanding chemical properties will be discussed. Al? integrations of p(r), K(r) and G(r) were performed using gaussian quadrature methods and yielded the to@ number of electrons correct 35
_.
Volume 8, number 1 -.
GHEMItiAL PHYSICS LETTERS-
to five.fi&ges (from five k&e orbital coefficients)- and kinetic energies correct to 0.0001 au.
REFERENCES
1 January 1911
:
[4] R..F. W.Bader and H. i.T. Preston; Intern. J. Quantum Chem. 3 (1969) 327.. 151P. Hohenberg and W.Kohn. Phys. Rev. 136 (1964) B864. -’ [6] R. G. Parr and J. E. Brown, J. ChemlPhys. 49 (1968) 4849. [7] B. Kelander, J. Chem.Phys. 51 (1969) 469. [8] .:6&. Allen and II. Shull. J. Chem.Phys. 35 (1961)
:
[I] k F.‘W. Bad&, P. M. BeddalI and P. B.Cade. J. :.4m. Chem. ~oc., to be published. [Z] A. G. Gaydon. Dissociation energies. (Chapman and Hall, London, 1968) p. 292.‘ [3] A. D. McLean and M. Yoshimine, Tables of linear molecular‘wave functions (IBM Special Publication, 1967)..
[9] J. M--Parks and Ft. G-Parr. (1958) 336. [IO] P. G..Lykos and R. G.Parr, (1956) 1166.
J.Chem. Phys. 28 J. Chem. Phys. 24
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._I I
i
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:
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._‘. _‘, ,- _., . . : ._:
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.__
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..
:-
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_‘_ _I
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:
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Volume 8, number 1
THE
SPATIAL
PARTITIONING
AND
MOLECULAR
1 January
TRANSFERABILITY
19il
OF
ENERGIES
R. F. W. BADER and P. M. BEDDALL Department
of Chemistry,
McMaster Received
University,
4 November
Hamilton,
Onfauio.
Canada
1970
This note examines the question of the transferabiiity of the energy of a hnded different a spatial
fragment between _._ molecules in terms of the transferability of the charge distribution. This is accomplished through partitioning of both the charge distribution and the energy of a molecular system;
1. PARTITIONING OF THE CHARGE DISTRIBUTIONS 9m
..a
9.m
We have recently proposed a partitioning and a classification scheme for molecular charge distributions, each scheme being directly related to observed (i.e., calculable) properties of the charge distribution [l]. The form of the charge distribution suggests a “natural” partitioning in the form of a plane P, perpendicular to the bond axis through the point on this axis at which p(r) attains its minimum value between a pair of adjacent nuclei. This procedure is illustrated in fig. la for the CO2 system. Integration of p(t) from infinity on the left up to the plane Pm defines the total electronic charge population on the left-hand oxygen nucleils, to. Integration of p(r) between the planes Pm and Pt; defines the total electronic charge population on the carbon nucleus tc . In addition, a further plane PA through a given nucleus A partitions a total population into bonded @A) and nonbonded (nA) contributions for a terminal nucleus, or into a number of bonded populations for a multiply onde centre. 8 8 ThusinCQ,tO=bO+ngandtc =bC+bC’ One of the‘ reasons for defining the above populations is that changes in a system caused by a change in the bonding environment are largely restricted to either the bonded or nonbonded regions. ThFs the principal changes in the charge Total molecular charge distribution for dO2: (a) constructed from a near H-F wave function [3] for COz, (b) constructed from the charge distribtitions of two $0 fragments using a wave function of comparable quality (3). The numbers denote the total populations (to and tC) as calculated in (a) and as predicted by:the fragment populations in (h) (see ?ahle.‘la). ,’ Fig. 1.
..
.-.
-:
_. ‘. : -_
I _
.
29
. ..
Volume-S.
number
-_ ..._ -_i--
1
CHEWCAL
----
PHYSICY
Pop&on
---._ ----
LETTERS
~Z&L3ons
1 ~Januarjr 1971
a)
.---Total.bonded populations
Bonded and nonbonded contributions A.
B.
ABC oco OW
“A 4.24 4.16
bA 4.76 4.76
b* B 2.00 2.03
bG %
bc
%
W%)
2.00 2.03
4.76 4.76
4.24 4.16
6.75 6.79
6.75 6.79
FCN
4.64
F$N
4.65
5.07 5.14
1.95 2.02
2.33 2.35
4.30 4.25
3.72 3.69
7.02 7.16
6.63 6.60
HCN H@N
0.43 0.51
0.62 0.66
2.65 2.60
2.31 2.35
4.28 4.25
3.71 3.69
3.27 3.26
6.59 6.60
WNO NqrO
3.65 3.57
2.96 3.43
3.91 3.43
3.lG 2.90
4.05 4.34
4.27 4.16
6.87 6.86
7.21 7.24
b(CC’)
h(C’H)
nA
bA
bc C’
6;s
0.45
0.63
bA C 2.76
bc’ C
HCCH b)
3.17
3.17
LiCCH
1.05
1.16
3.49
2.69
3.68
- ACC’H
nH 0.45
2.79
0.66
0.48
6.34
3.39
6.37
3.45
4.66
5.09
2.01
3.73
2.64
2.79
0.63
0.44
6.37
3.42
ClCCH
8.54
8.39
2.84
3.50
2.85
2.76
0.64
0.44
6.35
3.40
ABC
“A
b(AW
MBC)
4.23 8 .Ol
bA B 2.01 3.80
bc B
oco SC0
bA 4.76 7.17
FCCH
C.
2.76
bH 0.63
W’C)
b)
2.01 2.03
bC 4.76 4.77
6.77 10.97
a) All polyatomic wavefunctions in A and B are from ref.[3] and all are of near Hartree-Fock accuracy CICCH, The wavefunctions for CICCH and in particular for CO2 and SC0 of part C are of comparable (doubIe-zeta + polarizing functions). The diatomic functions are from P. E. Cade (see ref. [l]). b) Compare F-4 and H-q values with corresponding fragments in A.
distribution in terms of populations may be spatially isolated, and other populations undergo only minimal changes. One example of this is illustrated in fig. lb which shows a charge distribution for CO2 obtained by discarding the nonbonded population on carbon in two CO diatomic molecules in their XIC’ ground states and joining the two fragments at the resulting planar surfaces through the carbon nucleus. The populations, (see table la) both the totals and the bonded and nonbonded contributions, differ at the most by O.O8e-, between the actual molecular charge distribution (fig. la) and that constructed from the fragments (fig. lb); the total bonded pop lations between carbon and oxygen b(C0) = = b $! + bg differing by 0.04e’. The total population on-C in the diatomic species CO is very asymmetrically.partitioned between the bonded and nonbonded regions with nC, = 3.05 and bg : = 2.03. In addition, the nonbonded charge is in the_fckm_of a very diffuse distrection of relatively large spatial extent.-_In spit$of- the very different nature of the bonded ax@ nonbonded PO-. pulatiks, the .change in the populations bet$een CO and CO2 is almost entirely restricted to-the . .
6.77 6.50 except for accuracy
nonbonded fragment on carbon which in this case is set equal to zero (i.e. discarded). The small differences in the populations between CC2 and the CO fragments are surprising in the light of simple valency concepts which ascribe a triple bond and single pairs of unshared electrons to each nucleus in CO and double bonds and two unshared pairs to each oxygen in CO2. There are no changes in any of the populations which would correspond to such a gross transfer of charge from the bonded region of the diatomic CO to the nonbonded region of oxygen in the CO2 system. In table 9a a number of examples are given comparing the populations of ABC with the fragments A$3 and $C!, the vertical bar designating the fragment being transferred. In CQ, .FCN and HCN the total populations and the bbnded and
nonbonded contributions show little variation be tween the diatomic fragments and the polyatomic system. In NIT0 the totd populations do differ significantly between the two cases. .However, the total h&d&d populations b(q = bN, +bNY and &(Nq) = bN + 68 and the nonbonde %, popu. “I ations are little changed. Thus_ wh.en there-is a considerable‘ change @I the populations due to a’ ._ . . ..-_ .’ ., ..; _:. ._. .’
Volume 8, number 1
CHEMICAL PHYSICSLE’I’TERS
change in bonding, the change appears t-o be restricted to adjacent bonded populations. Thus charge is shifted from 6& - q and from bg - bg compared to the diatomic fragments, but the nonbonded and total bonded populations remain nearly unchanged. This shift of charge density wiflrin a. bonded region is further illustrated in table 1d:in which comparisons
are made between the populations
of a number of substituted acetylenes ACCH and the parent molecule. In these exam les the F +2 bonded population b (CC’) = bC’ differs by only O.O3e- over all four mo ecule!?‘but there is considerable change in the separate values of bc’ andbC from the symmetrical case of HCCE. ‘The dag in tables la and b indicate that the nonbonded and total bonded pop&atiozs are rather insensitive
to changes in bonding and may be
poor gauges of changes in properties. In none of the above systems is there any question of transferability of energy for example, in spite of the near constancy of the individual populations such as is found in CO and CO2. Close examination of figs. la and b shows that the charge distribution irt the bonded region of the ($0 fragment of the CO diatomic (fig. lb) is more contracted along the bond axis than it is in CO2. Thus the principal reason for the bond energy of CO being much greater than the average CO bond energy of CO2 is that the charge density is more contracted in the bonded region of the former compared to the latter. Charge populaticns (as defined here) undergo relatively small changes as the bonding environment is changed. The changes in properties
mining the properties of such Fragments. Thus we propose that the extent to which the AB rend
BC bonds of A2B and C2B are transferable to ABC is determined by the extent to which the charge distributions of the fragments A8 and I@ are identical, up to the plane PB in each fragment, to that found in ABC. The energy of OCS may be obtained additively from the energies of CO2 and CSZ to within a Eew kcal/mole [2],
E(OCS) = Q[E(CO2) +E(CS2)]
& 2 kcal/moIe
.
The populations of the CO fragments in OCO and and OCS are given table lc [3]. The changes in the bonded and nonbonded populations of the 60 fragment are small tid the total fragment popuLations differ by only O.O134e-, being largest in OCS. Fig. 2 compares the charge distributions of the $0 fragments in the two systems. in this case (unlike the case of the ($0 fragment in CO2 and CO, fig. 1) the differences in the distribution of charge as well as the populations are smalL.
caused by a change in the bonding
environment aye caused by a relatively smal2 redistribution of charge density within a given spatial region and not by a large cizmge in its population. A simple example of transferability of bond properties (bond additivity) is transferttiility of the AB and BC bonds in certain symmetrical systems ABA and CBC TVthe unsymmet:rical system ABC. The above results indicate that the planes through the nuclei represent at least partial barriers to the transfer of charge. A perturbation may be transmitted through the barrier causing a redistribution of charge within a bonded region (eg. $0 : TO) but very little change occurs in the total populations a.8 defined by these planes. These results suggest that transferability, when it does occur, should be reflected ln the iear or total constancy of the distri&&.ion of charge (not just the populations) in the fragments A$ and $C and that on65 could obtain a set of transfe*le properties b:y deter-
Fig. 2. Comparison of the ctige fragment
distribution of the 40 in SC0 (solid lines) with that in CO2 (dashed
lines}.
-31
Volume-8, number 1
CHEMICALPHYSICS.LZTTERS
It is impqrtant to note that the principal difference in p(r) for’ $0 b&ween CO2 and OCS occurs at t.& boundary plane in the outer diffuse regions of the Since p(r) must be single charge distribution. valued akd continuous at the boundary plane, such small changes in p(r) at the boundary will occur in general. They would be absent only if the radial distribution of the charge density on carbon in the. boundary plane was identical in both the a0 and $S fragments in their parent molecules. This situation is not obtained here (p(r) is more diffuse in $S than in $0 at the boundary) and is not expected in general. The question to be asked is whe!her near ‘ additivity of the energy is a result of a near transferability of the charge distributions ‘ of the fragments as defined by the plane PC; i.e., is theplzne PC the correct boundary surface for .the explanation of additivity ? To answer these questions& is necessary to define and determine the energy of a fragment Al& This is the subject of the following section. 2. PARTITIONING
OF THE ENERGY
To effect a spatial partitioning of the energy it is necessary to express the &ergy of the system as.a function of the three spatial coordinates defining the charge distribution, i.e. one must define an ,energy density. It is possible to write such an expression for the kinetic energy of a system and we have previously given definitions for two kinetic energy density distributions [4]. If the charge distribution is expressed in terms of the natural orbitals of the system: p(r)
= Cii$(r)
eibi(f) =Cf+)
&,d:
=Ur) -. z(A .‘
+ G(r)
,
(1)
.’ = - v+!rj
theorem to both sides of the
-.
form of eq. (1) with G(r) transposed
JK(r)dr-
JG(r)dr=JL(r)dr
(4)
,
we obtain
(5)
where d$/dz and dp/dn are the normal derivatives (outwardly directed) at the point dS on the surface S. Eqs. (5) and (6) are of course equal, the form& being written to illustrate that it applies separate11 to each natural orbital in the expression for p(r). If the boundary surface S is taken at infinity where both 9 and pand their normal derivatives are
Ai
,
January 1971
we obtain
the results
=0 andJK(r)dr
[4]
=IG(r)dr=F
.(7)
The
where ?-is the total kinetic energy of the system. However,. it was also demonstrated thatK(i) m2y be expressed in terms of the laplacian of p( r) and tbe._gradient+ of the orbital densities Pi(r) as [4]. .__
integrated
jL(r)dr
-' 'fK(r),dr = F ,
,+
Applying Green’s
zero
-then the mosi immediate definition of a kinetic energy densiiy is given by : K(r) = -+C Ai$i(r)V.2@i(r) , i for which one clearly has
:whtire
1
6) .
L(r)dr
= $$~~~podrrdr/~=~~~
,
(8)
where p(r) is expressed in cylindrical coordinates with .z lying alocg the internuclear aXis in the direction of negative 12&rough the plan?r boundary surface .and p!‘(z) is a one-dim‘ensionall representation of thetotal. charge. distribution, i.e., -p’(z) dz is-the total charge Iying betweek the two parallel planes at z and z + di Eq. (8) restits since dp[dn of eq. (6) c&be finite only in,thti perpendicular bouncltiy p&ie it nliciegs .B;: Th& the value .of .-
Volume 8. nu,nber 1
CHEMICALPHYSICSLETTERS
slope of the one-dimensional representation of the charge distribution p’(z) at the position of the boundary surface and its value for the adjoining fragment sharing the same boundary surface must be equal and opposite. Recognizing that Green’s theorem is a special case of Gauss’ theorem we are Led to an interpretation of the above equations through an identification of p(r), the charge distribution, as a scalar field. Thus -4L(r)
= V20(r)
is Poisson’s
= -4[K(r)
- G(r)]
I January 1971
However, the same conditions hold in another situation, one of direct consequence to the transferability question. Consider the transferability of the fragments A$ and BC from ABA and CBC to form ABC. Clearly, since in the symmetrical parent molecules A2B and C2B dp’(z)jdz = 0 at the position of the central nucleus, the integral of L(r) dr over the regions defining the fragments A$ and I$C will be zero. Thus in these cases we have &K(r)
dr = &
G(r) dr = T(AB) = $ i
;
equation and its solution yields
(11) s L(r)dr
=0
,
A@ that is, one will obtain one-half of the total kinetic energy. Furthermore, the virial theorem will apply inasmuch as where A = 1 r - r’/. Thus the total charge density at any point in space may be considered to be the scalar field arising from the source -4[K(tI - G(r)]. Application of Gauss’ theorem
to eq. (9) yields eqs. (41, (5) and (6) directly and
enables one to interpret dp/d?z as the vector field or gradient af the scalar field p(r), and the surface integrals involving dp/dn as yielding the net outflow of this vector field over the surface S. In general because of eq. (8), integration of G(r) dr or K(r) d r over some restricted volume of space will yield different values for the corresponding kinetic energy contribution for that region. Thus the question arises as to which function should be integrated to determine the kinetic energy of a given spatial region. Furthermore, we wish to make the assumption that for a system at internal equilibrium such that the virial theorem applies, we may equate the negative of a kinetic energy contribution from a given region of space, -FR, to ER, the contribution to the total energy from the same region. Let us consider both these questions, the choice of G(r) or K(r) for determining TR over a partial region of space, and the-equating of :FR with ER, in terms of the transferability problem. There are two situations in which we may answer both questions withcertainty. The first is the trivial one obtained when-the region of integration R is set equal to the whole of space. Inthiscase .-
$T = -$E = -E(A$)
.
(12)
The total energy of ABC will then be given by E(ABC) = E(AB) + E(@C)
.
For absolrcte tmnsfe?-Gbility to obtain, the
charge distribution of each fragment must remain unchanged in the construction oE ABC from ASB and C8B. Thus the integral of L(r) dr over the fragments A@ and $C in ABC must stiIL equal zero. (It is worthwhile here to recall tie theorem of Hohenberg and Kohn which states that for every charge distribution there is a unique value for the energy [5].) However, absolute transferability as defined by the conditions in eq. (11) will generally not be obtained as can be seen immediately from the property of the integral of L(r) dr as given by eq. (8). In ABC the integral of L(r) dr over either fragment-will deviate from zero as p’(zj will, in general, no longer b.2 a maximum at the position of the boundary plans. As discussed earlier in terms of the examples of CO and CS, even in cases where additivity is anticipated small changes in p(r) at the boundary pIa.ue are to be expected, as otherwise one is demanding an identical radial distribution of charge in the boundary plane for nucleus B in all of the systems of interest. Thus in general jk(r)dr
R
Cj- G(r)dr
R
for either fragment in ABC, the charge distribu-tion and the energy of each fragment is altered
and absolute transferability is not obtained. What we wish to demonsh%te is Mat even in tie :
:
-
._.
::
33 ‘_
: Volume 8, number 1
CtiEMXCALPHYSICS
absence of absolute transferability of p(t) an a.Imost exact additivity of the energy can still be obtained if. the smaII unavoidable changes in ~&,rf assume a special form. Additivity of the energy will be obtained if the values of either IB K(r) dr m &G(r) dr over both fragments remain unchanged from their vaiues in.the symmetrical parent species, since .:’ .$ &)dr
i- &K(r)dz’
A@
=
. _j-$(r)dr
+ &-G(r)dr
=
-E(AjW .
.If we make the assumption that &K(r) dr over a fragment may be identified with the kinetic enerG of the fragment, then the extent of additivity will ,be determined by the extent to which the change in Jd&(r) dr caused by the transfer
is absorbed
by a change in J”K(
r)dr rather than in &G(r) dr for both frag-
ments. The physical justification of this particuIar possibility as being the explanation of near additivity of the energy is as foliows. Parr and Brown [S] and later ‘Nelander f?] have shown that the virial theorem for a poiyatomic molecule may be written as
LETTERS
.1
Xn summary, for near additivity the small but necessary changes in p(r) caused by the transfer result in small changes in the forces acting on the nuclei and hence in the equilibrium bond lengths. The extent to which these changes, as measured by the deviation of [do’(z)/& 1from zero, are absorbed as changes in the kinetic energy and hence inlRK(f).ds fragment,
and
notdin &G(r)
< pAB(ABA).
Since
s K(r)dr A@
= FA$,
both this
I,( P) dr must decrease from’their Integral and s A$ vafue in ABA on transfer to ABC, and in particuIar $ L(r) dt < 0 in ABC. Since $ L(r) dr A@ A$ = $&) dr , it follows that JBCK(f) dr (=T;tac) , -: must incr&ke over its value .in CBC and hence
RBc(ABC) 84’ ..
< RBc(CBC)*. ‘.
’
dr for each
determines
the extent of additivityof the energy. Xg other words, the proposal is one which accounts for a nearly constant total energy or total kinetic energy on transfer (the sum of the integrated G(r) dr values} in spite of the small but necessary changes in the kinetic energy of each fragment separately. Other possibilities, such as changes in p(r) but not in &,&C(r) dr for each fragment would appear to be in violation of the theorem of Bohenberg and Xohn [5]. Substantial changes in the Integrated values of both K(r) dr and G(r) dr for each fragment (reflecting substantial changes In p(r) on transfer) which happen to cancel out on addition in ABC is unIikeIy and could be classed as an “accidental” additivity. We now give a numerical example illustrating the above proposals using the wave functions of McLean and Yoshimine [3] for the transfer of the $0 iragment between CO2 and SCO. The charge distribution
of $0 in CO2 yieIds
f K(r) dt = $$ G(r) $0 where the sum may be taken over all the bonds of the system [7]. The small adjustments in p(r) required at the boundary plane will result in an ,externsI virial contribution to the total kinetic energy of ABC from each bond. Thus in general RAB(ABA) f RAB(ABC) asRAB must change to a new equilibrium value in ABC. Consider the case RAB(ABC) > RAB(ABA). Then TAB(ABC)
January 1971
dr
= 93.7138 au
and $6 t(r)dr
= 0,
i.e., dp’(z)/dz = 0 at the boundary plane. In SC0 (as suggested by fig. 2) p’(z) slopes off towards the oxygen nucleus at the boundary plane, dp’(z)/dz is negative and hence = -0.0343
au or -21.5 kcal/moIe.
Ir’o L(i)dr We then correctly surmise that. R&SCO~ (=2.2016 a$? RcOfCO2) (= 2.1944 au) and that Z’@SCO) c Z@-J(CO~) as determined by . * This argument predicts that ia systems where one obtains near additivity of the energy, one bond length . must increase, the other mtist necessarily decrease on transfer of the fragments to ABC. This is SO for qnmple,
in the COgI CS&SCO
1’.
‘.
system. -.
Volume 8, number the integrals
CHEMICALPHYSICS LETTERS
1
of K(r) dr.
The value of
1 K(t)
dr
does indeed decrease in value on transfer (to 93.6774 au) and its decrease is equal to the decrease in s L(r) dr to within 1.5 kcal/‘mole. CO Thus the changes in the charge distribution of ($0 cn transfer as illustrated in fig. 2 are of such a nature as to decrease Ike kinelic mcqy of the system whiIe Ieaving the value of JGoG(r) dr almost unchanged, its value in SC0 being 93.7116 au, corresponding to a decrease of 1.5 kcal/mole in the energy of $0 on transfer. The value oE 1 L(r) dr in SC0 is + 0.0343 au SC indicating an increase in the kinetic energy of the Se fragment on transfer from CS2 to SCO. If this change is absorbed almost entirely by the change in the kinetic energy of the $C fragment on transfer (as found for the $0 fragment) one predicts an additivity of the total energy to within 2 or 3 kcal/mole in the COq,CS2-SC0 system. The standard (double) bond energy of the $0 fragment can be defined as E($O)
= E(0) + &F(C) + s G(r) dr
$0
where E(0) and E(C) are the energies of the ground state atoms with the integral evaluated for CO3. Other fragment properties could be defined, the dipole moment for example. Allen and Shull [8] were the first to point out that the electronic and nuclear repulsion energies of a system are not separately additive and that only through the use of the virial theorem, which relates the electronic kinetic energy to the total energy of the system can one provide a theoretical justification for additivity. Allen and Shull presented their arguments in terms of the transferability of the geminals defining an electron-pair wave function from one system to another with the assumption that the geminals could be so chosen as to be localized in the regions of the individual bonds. A wave function suitable from this point of view is the separated electron pair function proposed by Parks and Parr [9]. Lykos and Parr [‘i’], in discussing-the u-x separability problem have suggested that the total wave function may be expressed as a partially antisymmetrized product of two functions, these two functions themselves being antisymmetrized functions, one for the u electrons the other for 8. electrons. One can consider applyingthis concept of separability to the molecuIar
I Janun~ 1971
fragments as defined here by the boundary plane through a nucleus. Thus *(ABC)
= Jj,!/(A~>~($C)]
.
where p(A$) and $@C) are to be transferable antisymmetrized futctions for the A$ and $C fragments and the brackets denote a further partial antisymmetrization corresponding to the exchange of electrons between the two fragments. The present analysis. however; the transfer of such fragment wave
suggests
that
functiclns
between the different systems (all nt irttental eq:iilibrium) will in general not be possible. In order that the charge density and the wave function be single-valued and continuous at the boundary between fragments, changes in the wave function are necessary. (These changes may lead to a non-integral value for the electron population of a fragment.) Thus a necessary (but not sufficient) condition for absolute transferability of a fragment wave function between different systems is that the flux of Wp(r) for the fragment through the partitioning surface as defined by eqs. (5) or (6), be the same in all of the systems. En the case of linear systems this requirement is simply stated (and simple to test Ear) that dp’(z);‘dz at the boundary surface remain constant. As pointed out previously this is a stringent condition and one unlikely to be met in practice. Rather, what we have proposed from our considerations of the behaviour and properties of the charge distribution is a mechanism which acccunts foF the additivity of the energy between systems in spite of the small change in p(r) of each fragment which occurs on its transfer between systems. If the changes in the bonding environment of a fragment are remote, then the corresponding changes in its charge distribution on transfer couLd be vanishingly small and such favourable cases could approach absolute transfeiability. One could consider the possibility of preparing a fragment wave function for transfer by first placing the parent system in a non-equilibrium configuration. For example, the transfer of the shortened $0 fragment from a sLightLy asymmetrically distorted Co2 (to obtain a dp’(z)/dz < 0) to SCO. This and other questions raised by the present approach to transferability, together with more examples are to be reported. In addition, the usefulness of a spatial partitioning of the total kinetic energy in predicting and understanding chemical properties will be discussed. Al? integrations of p(r), K(r) and G(r) were performed using gaussian quadrature methods and yielded the to@ number of electrons correct 35
_.
Volume 8, number 1 -.
GHEMItiAL PHYSICS LETTERS-
to five.fi&ges (from five k&e orbital coefficients)- and kinetic energies correct to 0.0001 au.
REFERENCES
1 January 1911
:
[4] R..F. W.Bader and H. i.T. Preston; Intern. J. Quantum Chem. 3 (1969) 327.. 151P. Hohenberg and W.Kohn. Phys. Rev. 136 (1964) B864. -’ [6] R. G. Parr and J. E. Brown, J. ChemlPhys. 49 (1968) 4849. [7] B. Kelander, J. Chem.Phys. 51 (1969) 469. [8] .:6&. Allen and II. Shull. J. Chem.Phys. 35 (1961)
:
[I] k F.‘W. Bad&, P. M. BeddalI and P. B.Cade. J. :.4m. Chem. ~oc., to be published. [Z] A. G. Gaydon. Dissociation energies. (Chapman and Hall, London, 1968) p. 292.‘ [3] A. D. McLean and M. Yoshimine, Tables of linear molecular‘wave functions (IBM Special Publication, 1967)..
[9] J. M--Parks and Ft. G-Parr. (1958) 336. [IO] P. G..Lykos and R. G.Parr, (1956) 1166.
J.Chem. Phys. 28 J. Chem. Phys. 24
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._I I
i
-, :_I
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:
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.__
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