Energetic consideration of the vibrational potential function in the effective nuclear charge model—VIII.

Specrrochtmxa Prmted

I” Great

ACM, Vol 44A No

8 pp 809 817 1988 0

Brltam

0584-8539188 $3 oO+OCXl 1988 Pergamon Press plc

Energetic consideration of the vibrational potential function in the effective nuclear charge model-VIII. KEN

Dlvislon

of Chemistry,

Japan

Atomic

Energy

OHWADA

Research Japan

Institute,

Tokal-mura,

Naka-gun,

Ibarakl-ken,

(Received 9 October 1987, accepted 7 December 1987) hnear third-order differential equation for the dlatormc molecular potential-energy function, W,(R), ISformally derived through the dlfferentiatlon of the quantum mechanical virlal theorem with respect to the mternuclear distance, R The general solution of the equation IS derived and studled m detail for obtammg an adequate potential-energy function Smce the solution includes an unknown quantity, a reasonable assumption concerning such a quantity 1s made that leads to a useful, approximate potentialenergy function The approximate function thus obtamed is dpphed to the descriptions of molecular properties such as Iugher-order potential constants It IS dk0 shown that the above function 1s closely related to the model potential used m the effective nuclear charge model Lastly, electronic kinetic (T) and total potential energy (V) representatwe second-order &fferentlal equations are derived on the basis of the vlrlal theorem and solved Their general solutions are subject to investigations of the properties of the general solution for the third-order differential equation m questlon Abstract-A

1 INTRODUCTION

ation This IS described m Section 5 Section marizes the results obtamed m this study

6 sum-

In a prewous paper [ 11, we have discussed m detail the homogeneity properties of the molecular energies with respect to the atomic number At the same time, 2 FORMAL DERIVATION OF LINEAR THIRD-ORDER mhomogeneous linear second-, third-, fourth-, and nthDIFFERENTIAL EQUATION order dlfferentlal equations have been appropriately constructed from a linear first-order dlfferentlal equFor a dlatomlc molecule m the Born-Oppenheimer ation with a homogeneity constramt and solved for approximation, the quantum mechamcal vlrlal understandmg general potential forms of dlatomlc theorem [2] relates the total molecular energy W,(R) molecules However, the derlvatlons of such dlfferenof the nth electronic state and its first derivative to the tial equations are tentative dnd, therefore, more essennth electronic kinetic energy T,(R) as a function of the tial, theoretical derlvatlons are very desirable internuclear distance R In the present study, attention mainly focuses on the derlvatlon of a linear third-order differential equation RdK(R) (1) 7 + W,(R)= -T,(R), wlthout any constraint and the practical slgmficance of such a dlfferentlal equation 1s stressed through the T,(R) = (nlfln> (2) descrlptlon of molecular properties In Section 2, we attempt to derive the linear third-order dlfferentlal Here ? 1s the electronic kmetlc energy operator, and equation for the dlatomlc molecular potential-energy In) 1s the nth electromc state From the first derlvatlve function based upon the quantum mechamcal vlrlal of equation (1), theorem (2) In Section 3, the differential equation 1s Rd2W. d(nl%> solved to obtain the general solution (potential-energy 5+2$ (3) dR ’ function) Since the solution contams an unknown theoretical quantity which 1s analogous to a perturbaCLINTON [4] has first derived the mhomogeneous tion-type sum over states, a reasonable assumption linear second-order dlfferentlal equation for the dlaconcerning such a quantity 1s introduced that leads to tomic potential-energy function W, as follows a useful, approximate potential-energy function The approximate function thus obtained 1s applied to the (4) descrlptlons of molecular properties such as hlgherorder potential constants It 1s shown m Sectlon 4 that the approximate function IS closely related to the where Q” denotes the theoretlcal quantity analogous to a perturbation-type sum over states as given by model potential used m the effective nuclear charge model (3) Electromc kmetlc and total potential energy 4(4%>12 representative second-order dlfferentlal equations (5) Q,(R) = n m which are derived on the basis of the vlrlal theorem are The present section IS devoted to the derivation of a solved to further investigate the properties of the linear third-order dlfferentlal equation from the vlrlal general solution for the third-order differential equ-

m;n

w

_

w

810

KEN OHWADA

theorem Accordmg to the procedure by CLINTON[4], let us start from the second derivative of the vlrlal theorem d3W, d=W, R- dR3 +3-= dR=

(n@lnP)=

d2(nl?ln)

-

dR=

= -2{(n’l~ln’)+(nl~‘ln”)},

(6)

where In’) = dln)/dR and In”) = d21n)/dR2 The last expression in equation (6) 1s easily derived by, taking mto account that the kinetic energy operator T IS Hermittan and independent of R The problem 1s to calculate the two bracket terms of the nght-hand side m equation (6) For convenience, we intend to first calculate the term (n’l ?ln’) and next (nl ?ln”)

A Calculatton of the
Inserting equations (12) and (13) into equation (8) yields

.

..I

It has been shown by BROWN[S] that the In’) state may be expanded with the Im) states m the form of

(kl~lm)(nl~lm>(nl~r,V,lk> -L?. +c c k+“ltl*”

R=P’,-

WJ

(kl~lm)(nl~r,V,lk)(nl~r,V,lm) R= (14)

In equation (14), the second, third, and fourth terms may be further transformed to more tractable forms with the use of the followmg relationships ~~~

= -(nlfln>,

(15)

~~“(nlTr,V,Im>(ml~lk)

= --<4$lkh

(16)

In’> = C and

m#n



=m;”w-wn

m

Irn),

(7)

where fi = $+ 3 (P total potential energy operator) IS the adiabatic molecular Hamkoman, AIn> = W”ln), and al?/aR=Z? With equation (7), the term m question can be expressed as (n’lpln’)=$;”

(W-W)(W-W) n k

”

m which the former equation has already been proved by CLINTON[4], and proof of the latter 1s given m the Appendix I The result is that (~l~lm>(nl~l~>(nl$lm> R=(W-W,)(W-W I In)

‘“““““=&,;. + c

m

I
kfnR=(Wn-Wk)

To further develop equation (8), the two bracket termsof (klfi’ln) and (mlfi’ln) must be treated For this purpose, we have need of the followmg basic formulas [4] [r,V,, A] = -P,Z/m,+r,(V,P),

(9)

Cr,(V, P) = - P- R(aA/aR),

(10)

;nlQm>

= -(nlPlm),

+ c

I(nlQm>12

,,nR=W’n-

WA

+ (nlfln> R=

(17)

Now, since we have no need of dlstmgmshmg k from m m all the terms except for the first term m equation (17), tt IS easy to see that

(11)

which are easily venfiable Here, r, and V, denote the position and gradtent vectors, and P, and m, the momentum and mass of the tth particle From equations (9)-(1 l), we have immediately R(klfi’ln)

= -(nl?lk)+(W,-

which 1s the final result m the present subsection

W,) (12)

x <4Cr,V,lk), I = - (nl?lm)

+(W,-

x (4Cr,V,Im) I

term

In order to calculate this term, we have need of expanding the In”) state m the )k) states This 1s performed by BROWN’Smethods [S] as follows

and R(mll?‘ln)

B Calculation of the (nITIn”>

W,) (13)

In”> = c D,,lk), ken

(19)

Effective nuclear charge model-VIII

811

where D,, (n # k) IS given by

-

2(k(ii’jn)
(20)

The first term (I) m equation (24) IS further med by usmg equations (12) and (15) to

W&J’

l<~l~lW

Before proceedmg to the calculation of (nl fin”), let us consider the first term m equation (20) Noting equation (10) and dlfferentlatmg It one time with respect to R, we have

(I) = 2 1 lr+n R’(W,-

usmg

the

afi/aR =I?

Slmllarly, the second term (II) m equation (24) IS also transformed by usmg equations (12), (13), and (16) to

(~l~lk>(~l~lk>(~l~l~) (‘I) = 2,;.

g"= -$-A&{Sr,(V,P)}

(25)

R2

a2fi/aR2 = fi” and

abbrevratlons,

W,)

+ 2(nl%>

(21)

or,

transfor-

+2

(22)

Rz( W - W )( W - W )

“,$,

I(nlflk>l’ AR k+nR2(Wn-wh) +F’ ” k n c

m

(26)

where the last term AR IS given by the comphcated equation,

AR=-2x

(W,--

W,)(nl~Ik)(nl~lm)(mCr,V,Ik)

(W,-

~k)(~l~lk)(~l~r,V,lk)(~l~.r,V,l~)

C kfn

men

+2x

c

(27)

ktnmfn

wn-

In the derivation of equation (21), the relation of aQ/aR = afi/aR IS used With equation (22), the first term m equation (20) becomes

w,

The third and fourth terms in equation (24) no longer lead to tractable forms, and, therefore, they remam unchanged The sum of such terms IS abbreviated to

R2(nlflk)(klfi’ln)(nlfif[n)

AQ=-2x hfn

lw,-

wk)2

(28)

NW,-

By taktng mto account atlon (24) becomes

wk)

equations

(25)-(28),

Now, we can proceed to the calculation of the term m questlon From equation\ (19), (20), and (23), we obtain

(nl fin”)

=

+2c

kfnmfn

(4flkXklfi’)ln) -2

c

kfn

R(Wn-Wk)

kZnmfn

_2

c kfn

c

twn-

wk)(w”-

““k)(~~l~~~~~nl”‘ln) ”

k

(~l~lk>(~l~lk>(~l~l~) R2(K-%)(&-wm)

AR +-+z,

-

(6 ~lk)(kld’lm>(mlfirln~ +2c

c

w”,)

R’

AQ (29)

this bemg the desired result, which we were looking for, m this subsectlon Here, we return to equatron (6) and substitute the bracket terms of (n’( f(n’) and (n( ?I#) m the nght-

KEN OHWADA

812

hand side wtth equations (18) and (29) Then, we have Rd’ Wn

d2W,

dR3

dR2

-+3-=

followmg relationship must hold

Q,(R) = Q:,(R),

(35)

from which we have

AR +Rt+s

AQ

1

(30)

Remembering equations (1) and (4), equation (30) may be rewritten as follows

J&%+9@%+ 18Rz+6Wn=Pn(R),

This means that the perturbation-type sum over states can be replaced by the derlvatlve of the electromc kinetic energy In a similar way, the linear third-order differential equation m question can be obtained by taking R times equation (33), subsequently dlfferentlatmg It with respect to R and finally adding the resultant to twice equation (33)

R3f$+9R’ft$+18Rz+6Wn

= P;(R),

(31)

(37)

1

where P,, IS given by P,(R)=

-2

3 1

c

(4 %X4

k#nm#n

+AR+AQ

tw,-

flO(4 wk)(wn-

flm) wm)

1

Thus, we have formally obtained the mhomogeneous, linear third-order differential equation for the dlatomlc molecular potential-energy function from the second-order denvatlve of the quantum mechamcal vlrlal theorem with respect to the internuclear distance However, it 1s necessary to mvestlgate in detad the physical propertles of the Pn term m equation (32) This problem will remam to be solved m the future

3 APPROXIMATE

POTENTIAL

ENERGY FUNCTIONS

Before proceedmg to a detaded dlscusslon of potential energy functions obtained by solving the linear third-order dlfferentlal equation (equation (31)), we digress at this point to search for another convement way of derlvmg the dlfferentlal equation m question Let us agam consider the first-order differential form of the vlrlal theorem as already given m Section 2 If we take R times the first derlvatlve with respect to R of equation (l), and add it to twice equation (l), we immediately have (33)

Q;(R)= -; $[R~T.(R)~,

(34)

of which the dlfferentlal form Just corresponds to equation (4) derived by CLINTON [4] Of course, the

It 1s found at equation (37) equation (31) cated formula ie

(38)

first sight that the dlfferentlal form of 1s completely ldentlcal with that of Therefore, we may equate the comphP,(R) (equation (32)) to equatron (38), (39)

P,(R) = K(R)

It goes wlthout saying that the methods described above are easily extended to the problems searching for the appropriate higher-order dlfferentlal equations above the third-order one if necessary Now, we return to the problem of understandmg potential energy functions for dlatomlc molecules Since equation (31) or (37) 1s the hnear dlfferentlal equation, we can easily solve it by the usual manner The general solution 1s given by

W”=;{Cl+;IPn(R)dR}

RP,(R)dR

which 1s exact and contams three mtegratlon constants, C,, C,, and C, As 1s well known, such constants may be determined by imposing three boundary condltlons If we take the followmg condltlons at the equlhbrlum position R = R, (1) W,(R,)

= WE (dlssoaatlon

(2) (a W,/dR), (3) (a2 W,/aR2),

energy),

= 0 (eqmhbrmm condltlon), = K (quadratlc force constant),

(41)

Effective nuclear charge model-VIII equation

(40) becomes

813

and (3W:+KR2/2)1

w, = ;{

3W:R+KR”,2)+$

V,(R) = V,o+

P”(R.)dR.)

R

(W;+KR2/2)R3 -

R’

’ (45)

R

R’P,(R’)dR’ s

ii

(42)

This equation ~111 completely determme the exact potential energy W, for a dlatomlc molecule if knowledge of P,(R), Wz, K, and i? are avallable Howeveras already shown m the previous section, the P,(R) term m equation (42) 1s expressed as the comphcated equation (equation (32)) and, therefore, we have no means of evaluating the exact quantltles of P,(R) at present Here, we intend to seek approximate potential energy forms under a reasonable assumption for P,,(R) The forms we choose ought to give predlctlons which agree reasonably well with experimental observations For this purpose, we use the P,(R) = const asyumptlon for R near i?, of which the appropriateness has already been dlscussed m detad m the previous paper [l] Replacing the P,(R) = const assumption mto equation (42) yields w = w,+(3W;R+KR3/2) n n R

= L = _ 6( W,e + 3KR2r2) R3

R2 KR5/2)

R3

. ’

(43)

where Wt denotes the charactenstlc constant of the molecule and 1s independent of R Equation (43) 1s one of the desired results m the present section It should be remarked that WE m equation (43) no longer expresses the exact dlssoclatlon energy because of mtroductlon of the P,(R) = const assumption This pomt will be discussed later There 1s no doubt that the approximate potential energy function (equation (43)) thus obtained 1s useful for simple descrlptlons of molecular properties For example, we can easily evaluate the R-dependent molecular energy components such as the electronic kmetlc energy T,(R) and the total potential energy V,,(R) mcludmg the nuclear-nuclear repulsion energy With the aid of the vlrlal theorem (equation (l)), the kinetic and the total potential energies are respectively given by (3 We+ KR2)R2 ” R2

2( W: + KR2/2)R3 +

R3

(46)

(3W;R2+KR4)

+ (WEE3 +

T,(R) = T,o-

where Ti = - Wi and Vf = 2 Wz Here, it 1s remarkdble that the linear coefficient m W,(R) does not enter T,(R) and that the quadratic coefficient m W,(R) does not enter V,(R) The other energy components such as the electron+zlectron repulsion energy, the electronnuclear attractlon energy and the Hartree-Fock elgenvalue sum [6] will also be obtained from equation (43) by making use of the HellmannPFeynman theorem [7] as well as the Hartree-Fock formula [6] within the framework of homogeneity assumptions of the molecular energy [ 1, S] We may now expect that equations (44) and (45) will give better predictions than those [9] derived from the solution of the linear second-order dlfferentlal equation (equation (4) or (33)) This may be supported by systematic studies of higher-order potential constants for various dlatomlc molecules The remainder of this section will be concerned with calculations of such potential constants Let us start from equation (43) Dlfferentlatlon of It wrth respect to R yields expressions for the potential constants,

-’

(44)

=M=

= o =

_

’

(47)

72( W,e + Kk?‘) R4

’

(48)

240(3 WE + 5Kk?‘/2)

85

1 (49)

where the subscripts e denote that the dlfferentlals with respect to R are evaluated at the equlhbrmm posltlon, and K, L, M, are the potential constants of quadratic, cubic, quartlc, As seen from equations (47H49), all the values of potential constants will be calculated if the quantltles of W$ K, and-l? are. known for a molecule However, we encouiter a dlfii- ’ culty m how to obtain the Wi value which cannot be taken as the experimental dlssoclatlon energy as already stated m the preceding section This arises from the mtroductlon of the P,(R) = const assumption To overcome this difficulty, let us define the quantity W,‘/KR2 = C If we treat C as an adjustable parameter and choose It to fit L and M values m an average way, then the evaluation of equations (47H49) will be easy The experlmental values of K, L, are obtained by normal coordinate analyses M, 0, with the use of vlbratlonal data

KEN OHWADA

814 Table 1

Cubic (L) and quark

potential state dlatomlc

Molecule Hz L1, C,

L

G G G A%,

Calc

-20 -63 -149 -192 -139

-21 -59 -144 -158 -132 -151

94 259 989 582 879

91 313 1060 992 963 1174 1458 326 1669 371 651 475 396 820 345 194 397 690 1986 1467 912 792

Na,

G G

-188 -78

P,

G

OH

G

-225 -58 - 102 -81 -64 -127 -55 -36 -62 -105 -264 -191

HF LIH AIH CO

G G A2A G G G G

NO

A’77 G

BO

-155

G

-131 -104

A%,

ct

Obs

0,

dnd excited

M Calc

G a’ng

A%’

for ground

Obs

Nz

HCI CH

constants(M) molecules *

-184 -69 -215 -61 -97 -74 -62 -121 -55 -36 -61 -99 - 249 -186 -127 -112

1120 1410 210

1540 407 591 385 367 737 341 183 383 610 1806 1400 855 877

-0 6778 -0 5974 -0 1904 -0 4484 -02217 -0 0675 -00261 -06691 -0 0723 -0 4841 -0 3624 -04161 -0 4204 -0 3372 -04417 -0 5721 -0 3994 -0 2952 -00075 -0 0292 -02408 -0 2663

*All quantmes m reduced umts G Ground state t Flttmg parameter, C= W:/Kt?

The results obtained thus are shown m Table 1, m which expenmental results [9, lo] are also mcluded for comparison As seen from Table 1, the agreement between the calculated and experimental results 1s favorable, and this p;lves support to the relatively high accuracies of equations (43)-(45) Another Interesting of equations IS to calculate the ratio of potential constants, PCR, PCR =(KM/LZ)1’2,

(50)

and to compare it with experimental or theoretlcal results For PCR of equation (50), FROST and MUSULIN [ 1 I] gave an average experimental value of 086 On the other hand, PARR and co-worker [9] obtained an approximate theoretical value of 1 from conslderatlon of a general solution derived from the linear second-order differential equation, and also obtained an empirical value of 0 89 using a simple Morse function It 1stherefore suggested that the PCR value m question lies m the range of 0 86 to 1 Substitutmg equations (46)-(48) into equation (50), we have PCR={2KR2(W,‘+Ka2)}“z IW,e+3KR2/21

’

(51)

constderatlon of a general solution for the linear second-order differential equation, the PCR value becomes 1 m accordance with the result by PARR et al c91

4 ON THE POTENTIAL EFFECTIVE

ENERGY FUNCTION

NUCLEAR

In this section, it IS shown that the potential energy function W,(R) derived from the general solution of the linear third-order dlfferentlal equation 1s closely related to the model potential used m the ENC model [3] As already stated m the preceding section, the general solution of the third-order differential equation (equation (31) or (37)) 1s given by equation (40), and includes three constants of integration, C,, Cz and C, Here, let us consider how to mcorporate the effective nuclear charges defined m the ENC model mto the above integration constants This may be accomplished by replaang the third condltlon m equation (41) with the followmg one

a2w,

( > aR2

from which we may evaluate the PCR values for the present case The calculated results are shown m Table 2 The average value 1s0 968 with a standard deviation of 0028 and 1s within the criterion described above This also assures the high accuracies of equations (43)-(45) It goes without saying that if we set Wi/Ki?* = C = - l/2 of which the value may be obtained from

USED IN THE

CHARGE (ENC) MODEL

e=+

2z*z*

(52)

which can be derived by the apphcatlon of secondorder perturbation theory to dlatomlc molecules [ 121 Here, Z*‘s are of course the effective nuclear charge defined m the ENC model [3] By the use of the first and second conditions m equation (41) as well as equation (52), we obtain the

Effective nuclear

Table 2 The calculated

potential

State

PCR

G G G G G G G G G G G G G G G

0 969 0994 0971 0976 0946 0979 0944 0 954 0916 0912 0999 0 996 0997 0 998 0996

Molecule “2 LIZ C* N, 0, Na, S], P, S, Cl, OH “Cl CH HF LI”

charge

constant

model-VIII

815

ratlo (PCR)* of dlatomlc State

Molecule

BO CO

CH OH

following

see equation

0 995 0 904 0 985 0 947 0 978 0 998 0953 0963 0981 0 943 0 928 0951 0 990 0 982 0 992 Average = 0 968 devlatlon = 0 028

(50) m the text

equation

solution

PCR

G G G G G A% g I ;j: /l%c: A’77 u’%+ u%, A’A czc+ A%+

Al” EN co NO BO c, N,

Standard * PCR =(KM/f.*)“*, tG Ground state

(54) may be rewritten W,(R)=

(3 w,eR+ 2Z:Z;)R R2 +(w;R+z:z:)R’ R3 where W,” denotes the characterlstlc constant molecule and IS Independent of R By takmg the followmg abbrevlatlons,

(54) of the

N,=(l+:S).

-(2+ZJ,

N3=(1+5$),

(55)

N,Z:Z*R’ ,,/

N,Z,*Z;i? R2 (56)

’

which IS none other than the potential energy function used m the ENC model [3] Thus, we can derive the model potential of the ENC model from the lmear third-order dlfferentlal equation under the reasonable assumption of P,,(R) = const for R near l? The foregomg interpretation may be extended to the case of a hnear nth-order dlfferentlal equation From conslderatlon of Its general solution, we ~111have the complete model potential of the ENC model

5 MOLECULAR ENERGY COMPONENTS In Section 3, we have derived two simple expresslons for the moiecular energy components of the electromc kmetlc T,,(R) and the total potential energies V,(R) from the approximate W,(R) usmg the vlrlal theorem, and support mdlrectly their accuracies through the calculation of higher-order potential constants The present sectlon IS devoted to the formal derlvatlon of such energy components from somewhat different pomts of view and to detalled dlscusslons of their functlonal forms By comparmg equations (6) and (30), we obtam

d2(nlfln) Iv,=

m the form of

N,Z:Z* W.” f------J+ R +

Agam, mtroducmg the P,(R) =const assumption, as already dlscussed m Section 3, mto equation (53), we have the simple, approximate potential energy functlon, (3 W,eR + z:z*) W,(R) = W,” f---L R

molecules

dR2

I %;ln)

=6 c ~Wlm)12

mt.R2(K-

P,(R) +R2 ’

&,I (57)

816

KEN OHWADA

where P,(R) 1s given by equation (32) Notmg that

for R near the eqmhbrmm posltlon as has been done m the previous sections, and to seek approximate forms for T. and V, Doing this, we have from equations (63) and (6%

equation (57) 1s easdy transformed to d2(nlfln)

R2

+ 6Rd(nl~ln) dR+6
dR2

(65)

V”(1) V”(2) V,(R)= V,(O)+ R + R3

(66)

and

or R2d2 T,(R) + 6Rd7,(R) do + 6T,(R) = P,(R), dR=

T,(l) T.(2) T,(R)= T.(O)+ F +R3

(60)

where T,(z) and V,(z) (z =0, 1, 2) are constants characteristlc of the molecule At first sight of equations (65) where, of course, T,(R) = (nlfln) (equation (2)) and (66), we find that their forms are compatible with Equation (59) or (60) 1s the mhomogeneous, linear equations (44) and (45) as already denved m Section 3 second-order dlfferentlal equation which IS derivable It 1salso mterestmg to compare the functional forms of from the quantum-mechanical vlrlal theorem Since equations (65) and (66) with those obtamed from the this equation 1s the kinetic energy representation, we lmear second-order dlfferentlal equation (equation call It the T-representative differential equation The (33)) by PARR, et al [9] samedifferential form as equation (60) can be obtained Lastly, let us show by a parameter fittmg technique directly from equations (38) and (39) that the functional forms of equations (65) and (66) are Slmdarly, we can derive the mhomogeneous, linear good representations of the behaviors near R In Table second-order dlfferentlal equation for the total poten3, calculated and Hartree-Fock values [13] of the teal energy (V-representation) as kmetlc energy are given for the ground state hthmm hydride molecule In Table 4, calculated and R2d2 + 5Rd(nl fin> ----++(nlPln)= -P,(R), Hartree-Fock values [13] of the potential energy are dR= dR also given for the same molecule As seen from these (61) tables, the present, direct tests reveal that equations or (65) and (66) well reproduce the R-dependent kinetic R$2V,(R)+ 5RdV,(R) (62) and potential energies even for relatively large R F + 3 V,(R) = -P,(R), dR2 where V,(R) = (nl Pin) In the derivation of equation (61), we have used two equations (4) and (60) The dlfferentlal equations of equations (60) and (62) are easily solved by the usual manner and therr general solutions can be wrltten down as

6 SUMMARY

The results obtained m this study are the followmg (1) An mhomogeneous, lmear third-order dlfferentlal equation for the dlatomlc molecular potential energy

Table 3 Calculated and Hartree-Fock values of electromc kmetlc energies for ground state hthmm hydride

(63) +A{C,z +SR”P.UWR},

R*

and G(R)=;{

C,, +$‘dR)dR)

+;{

Cvz -;jR’Pn(R)dR},

(64)

where c’s are the integration constants It 1s remarkable that the l/R term does not enter TH and that the l/R2 term does not enter V. Also, it IS easdy verifiable that the sum of equations (63) and (64), 1e W, = T. + V., IS ldentlcal with the general solution of the hnear third-order differential equation (compare with equatlon (40)) From this point of view, we need to consider the properties of equations (63) and (64) If the function P,(R) m equations (63) and (64) was exactly known, the solutions of T, and V, would be lmmedlate This 1s too ambltlous Here, we mtend to introduce the reasonable assumption of P,(R) = const

200 220 240 260 280 290 3 0156 3 034i 3 10 3 20 340 3 60 3 80 400 420

Kinetic energy? Calculated Hartree-Fock$ 8 38701 8 26436 8 17108 8 09848 8 04088 801644 7 991316 7 987388 7 97447 7 95638 7 92486 7 89844

8 31512 823109 8 15758 8 09431 804031 8 01644 7 991316 7 987388 7 97423 7 95563 7 92282 7 89518

7 87609 7 85700 7 84057

7 87196 7 85252 7 83631

*Internuclear distances In Bohr umts THartree umts are employed $.SeeReference [13] §Expenmental i? value TCalculated j? value

[Error1 007189 0 03327 001350 000417 000057 0 0 0 0 00024 0 00075 0 00204 0 00326 000413 000448 0 00426

Effective nuclear

Tdble 4 Cdlculated and Hartree-Fock values of electrostatic potentlal energies for ground state hthmm hydride Potentldl R*

CdlCUldted

200 2 20 240 2 60 2 80 2 90 30154 3 034’1 3 10 3 20 3 40 3 60 3 80 400 420

- 16 29223 -1620753 -16 13696 -1607724 - 16 02605 -1600310 - 15 97863 ~ 15 97470 -1596165 - 15 94287 - 15 90862 -1587817 - 15 85093 -1582642 - 15 80424

energyt Hartree-Fockj

0 06389 0 03050 001275 0 00406 0 00057 0 0 0 0 00025 0 00079 0 00220 0 00367 0 00488 0 00566 0 00594

-1622834 -16 17703 -16 12421 - 1607318 - I6 02548 - 1600310 -1597863 - 15 97470 -1596140 - 15 94208 - 15 90642 -1587450 - 15 84605 ~ 15 82076 -15 79830

(16) m the text, let us consider

(A-l)

mechamcal

S,=i.l~.v.ilk:i.lt.,V~n~~~,~,~~ Slmllarly,

we obtam

, equation

C n CV,r, m (mlflk) I> rnf” i:i I

Here, by makmg use of the quantum equation (A-l) IS reduced to

from equation

S,=(n/fV.r.jlk)+i:n/ZI,V,ln)(n,hk)

, Now, remembermg that [r,, ?] = IPJm, atIon (A-5b) becomes by the cham rule

(A-5b)

and V, = rP,, equ-

= -(n,fik)+;{(n,(Z?-2i) +(2%2?)+

Ik)} (A-6)

C(V,r,+r,V,)P=

APPENDIX I

S,,=

(A-3) and (A-4) yields

= - (nl ?lk),

energies

[r, V] =const

The sum of equations

which Just corresponds to equation (16) m the text Another proof IS given as follows Notmg the operator relation that

function has been formally derived from the second derlvatlve of the quantum mechanical vlrlal theorem with respect to the Internuclear distance and solved to obtam the approximate potential energy form of dlatomlc molecules (2) The approximate potential energy function thus obtamed has been successfully apphed to the calculatlons of higher-order potential constants and molecular energy components (3) It has been shown that the above potential energy function 1s closely related to the model potenteal used m the ENC model (4) Inhomogeneous, hnear second-order dlfferentlal equations of the electromc kmetlc (T-) and the total potential energy (V-) representations have also been derived on the basis of the vlrlal theorem and solved to discuss the functlonal forms of kmetlc and potential

By taking into account that can also be rewntten as

817

model&VIII

[Error1

*Internuclear dlstanLes m Bohr umts t Hartree umts are employed $See Reference 1131 QExperlmental i? value 4Calculdted l? vdlue

In order to prove equation the sum,

charge

(A-l)

(A-2) sum rule,

(A-3) (A-2), (A-4)

which IS easdy venfiable, form of

- ?~(V,r,+r,V,),

equation

(A-5a) IS rewritten

S,, = -1 Now, we can obtam

(A-7)

from equations

= - (nl?lk)

,

m the

64-8)

(A-5a) and (A-8)

(A-9)

REFERENCES

[l] K OHWADA, J them Phys 87, 4727 (1987) [2] J C SLATER, J them Phys 1, 687 (1933), Quantum Theory of Molecules and Sohds, Vol 1 McGraw-HII1 New ?o;k, 1963 c31 K OHWADA, J them Phys 72,1(1980), 76,2565 (1982) M W L CLINTON, J them Phys 36,556 (1962), 38,2339 (1963) c51 W B BROWN, Proc Cambridge Pi111 Sot 54, 251 (1958) C61 D R HARTREE,Proc CambrIdge Phrl Sot 24,89, 111 (1928), V FOCK, Z Physlk 61, 126 (1930) c71 H HELLMANN,Emfuhrung m die Quantenchenue Franz Deutlcke, Lelpzlg (1937), R P FEYNMAN, Phys Rev 56, 340 (1939) PI R G PARR and S R GADRE, J them Phys 72, 3669 (1980) c91 R G PARRand R F BORKMAN,J them Phys 46,3683 (1967), R F BORKMAN and R G PARR, tbrd 48, 1116 (1968) Cl01 G HERZBERG,Molecular Spectra and Molecular Structure I Spectra of Dlatonuc Molecules D Van Nostrand, New York (1950), E R LIPPINCOTT,D STEELE and P CALDWELL, J them Phys 35, 123 (1961), D STEELEand E R LIPPINCOTT,lbrd 35, 2065 (1961) Cl11 A A FROST and B MUSULI~, J Am them Sot 76, 2045 (1954) Cl21 J N MURRELL, J molec Spectrosc 4, 446 (1960) Cl31 P E CADE and W H HUO, J them Phys 47, 614 (1967)