Application of potential constants: Empirical determination of molecular energy components for diatomic molecules

Spectrochimica Acta, Vol. 45A, No. 4, pp. 487 490, 1989.

0584 8539/89 $3.00+0.00 © 1989 Pergamon Press plc

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Application of potential constants: Empirical determination of molecular energy components for diatomic molecules KEN OHWADA Division of Chemistry,Japan Atomic Energy Research Institute, 319-l l, Tokai-mura, Naka-gun, Ibararakiken, Japan

(Received 24 May 1988; accepted 9 November 1988) Abstract---The potential (force) constant of a diatomic molecule is applied to determine the molecular energy components such as the electronic kinetic(T) and electrostatic potential (V) energies.The theoretical framework of the method is constructed from T and V representations of the quantum mechanical virial theorem. To confirm the utility of the method developed here, the calculated molecular energy components of diatomic molecules are compared with available Hartree-Fock data, It is concluded that the present method is simple and powerful for evaluating the molecular energy components of various diatomic molecules. 1. INTRODUCTION

nic kinetic energy T.(R) as a function of the internuclear distance R:

In studies of molecular force fields, one of the most significant problems is to determine the exact general valence force constants such as bond stretching, angle deformation, bond-bond interaction, angle-angle interaction, and bond-angle interaction in polyatomic molecules. This problem has been approached from various points of view by many investigators, and sufficient data on it now exists. The next problem is to make the best use of these potential constants for an understanding of physical and chemical properties of molecules. In the present study, we take up, as a first step, the problem of determining empirically the molecular energy components such as the electronic kinetic and the total electrostatic potential energies for diatomic molecules, and try to solve this problem. In Section 2, we describe the T and V representative first order differential equations [1] of the quantum mechanical virial theorem [2], which provide some basic equations for predicting the molecular energy components in question. In Section 3, an inhomogeneous linear second order differential equation [3, 4] is constructed from the solutions of the T and V representative first order differential equations and used to establish the potential constant-energy relationships. In Section 4, the accuracy of the method developed in Section 3 is tested in comparison with available Hartree-Fock data [5] and discussed in some detail.

g ( d W . / d R ) + IV. = - T.,

(1)

W. = W.(R) = ( nlHIn >,

(2)

7". = T.(R) =
(3)

Here, H and T are the molecular Hamiltonian and the electronic kinetic energy operator, and In > is the nth electronic state. The nuclear repulsion term can always be added to W.(R) since Eqn (1) is invariant to addition to the left-hand side of a term in 1/R. By taking into account that H = T + V (V = potential energy operator), Eqn (1) can be rewritten as R(d T./dR) + 2 7". = - [R(d V./dR) + V. 3,

(4)

where V. is the potential energy of the nth electronic state given by

V. = V.(R)= < nl VIn ).

(5)

Here, if we set an unknown function Q.(R) to Eqn (4), we immediately have

R ( d T J d R ) + 2T. = Q.(R)

(6)

R ( d V . / d n ) + 1I. = - Q.(R).

(7)

and

The Q.(R) function in Eqns (6) and (7) is formally derivable from the calculation of the derivative of (nl T i n ) with respect to R, i.e. d ( n l T l n ) / d R = 2(nl Tin'),

(8)

where In'> = din >/dR. The calculation of Eqn (8) may be performed by expanding In'> [3] as

2. T AND VREPRESENTATIONSOF THE QUANTUM MECHANICALVIRIALTHEOREM We start by giving some basic equations which yield approximate relationships for calculating the molecular energy components in question. For the sake of simplicity, let us consider a diatomic molecule in the Born-Oppenheimer approximation. The quantum mechanical virial theorem [2] relates the total electronic energy W.(R) and its first derivative, to the electro487

In') = ~ ( m # . ) ( m l d H / d R I n ) l m ) / ( W . - W.,). (9) According to the procedure of CLINTON [4], Eqn (8) then becomes

d(nlTln)/dR

=

- ~,(m..)21( nl Tlm )12/[ R( W. - Wm)] --2(nlTln>/R,

(10)

488

KEN OHWADA

from which, using - (hi VIm ), we have

the

relation

(nlTIm)=

Noting the following relationship,

d IV./dR = - (I/R 2) [Co - ~Q.(R) dR]

Q.(R) = - ~ c . , . ) 2 ] ( n ] Tlm )12/( W. - Wra) = -~t,...)2[(nlVlm)12/(W.-W,.).

-(2/Ra)[c,+~RQ.(R)dR], (11)

Substitution of Eqn (11) into Eqns (6) and (7) yields the T and V representations of the virial theorem [1] which we were looking for. Equations (6) and (7) are inhomogeneous linear first order differential equations. Their general solutions are given by

T.(R)=(1/R2)[C,+JRQ.(R)dR]

(12)

V.(R)=(1/R)[C~-~Q.(R)dR],

(13)

and

where Ct and C~ are the constants of integration. Introducing the drastic assumption [6] of Q,,(R) = const for R near the equilibrium position, Eqns (12) and (13) can be reduced to

T.(R) = To+

T,/R 2

(14)

and

V.(R) = Vo + V,/R,

(15)

which just correspond to the functional forms extensively studied by PARR and BORKMAN [6]. Here To, T~, Vo and V~ are constants characteristic of the molecule and will be determined empirically in the next section.

3. DERIVATION OF FORCE CONSTANT EXPRESSIONS

In this section, we derive force constant relationships appropriate for evaluating the molecular constants such as To, 1"1, Vo and V1 in Eqns (14) and (15). This problem may be solved by constructing an inhomogeneous, linear second order differential equation. The total molecular energy W.(R) is given by the sum of Eqns (12) and (13):

W,,(R) = V,,(R)+ T.(R) = (1/R)[Co-~Q.(R)dR] +(1/R2)[C,+~RQ.(R)dR].

we can easily find that Eqn (18) leads to the second order differential equation in question: R2(d: IV./dR 2) + 4g(d W./dR) + 2 W. = - Q.(R), (20)

Q.(R)=(1/R){(d/dR)[R2T~(R)]}.

Taking R 2 times the second derivative with respect to R of Eqn (16), we have

(21)

Equation (20) seems uniquely appropriate and has been first derived and discussed by CLINTON [41, subsequently by PARR and BORKMAN [61. Equation (21) is equivalent to Eqn (6). Now, we can obtain the force constant expression under consideration. Evaluating Eqn (20) at the ground state equilibrium configuration, we have

R2Ke + 2 We = - [Q.(R)]e,

(22)

where K e = (d 2 WJdR2). is the quadratic force constant, We = W,,(R,) the equilibrium molecular energy and R e the equilibrium internuclear distance. Here, if we employ the Q.(R)= const assumption [6] in the preceding section, i.e. use Eqn (14) as well as Eqn (21), the term of [Q.(R)], in the right-hand side of Eqn (22) may be reduced to [Q.(R)], = 2To,

(23)

where To is the R-independent constant in the kinetic energy expression (14). From Eqns (22) and (23), we have ~ useful formula,

R2Ke = - - 2 W . - 2 T o = 2(W~ - W¢)- 2(W~ + To) = 2 0 , - 2(W~ + To).

(24)

Here D e ( = Wo~-- IV,) denotes the dissociation energy, and W~ is the molecular energy at R = oo. Provided that the diatomic molecule (AB) dissociates into neutral atoms (A and B) at R = oo, W~ may be expressed as the sum of electronic binding energies of free atoms, W~ = WA+ W~.

(16)

(19)

(25)

Thus, the R-independent TO can be determined accurately from Eqn (24) by the requirement of neutral dissociation products (Eqn (25)),

~Q.(R)d R ]

TO= --(WA+ WB)+D,-(1/2)R2Ke,

--(6/R 2) [C, + ~RQ.(R) dR] = - Q.(R).

since all the quantities in the right-hand side are obtained experimentally. The other molecular constants of T~, Vo and V~ in Eqns (14) and (15) may be evaluated from both the virial theorem (Eqn (1)) and the equilibrium condition under the Q.(R)= const assumption as

R2(d 2 W./dR 2)-(2/R) [Co -

(17) This is further transformed to g2(d 2 W J d g 2) - 4R{(1/R 2)[Co -

~Q.(R)dR]

(26)

+(2/R 3) [Ct + ~RQ.(R) dR]} + 2 {(I/R)[Co-~Q.(R) dR]

+ (1/R 2) [C, + ~RQ.(R) dR] } = - Q.(R). (18)

Ta = (1/2)R~ K.,

(27)

Vo = - 2 T o - 2 ( W A + WB)-2De+R2K.,

(28)

I,'1 = -- R~ K e.

(29)

Molecular energy components As seen from the above arguments, the electronic kinetic as well as the total electrostatic potential energies for diatomic molecules are easily calculated at any internuclear distance near the equilibrium position with the aid of experimental values (R e, K , , D c, WA, WB). The m e t h o d described a b o v e m a y be extended to the cases of polyatomic molecules by the use of the extended virial theorem [7]. This will be reported in the near future.

489

In order to test the accuracy of the m e t h o d described in the previous section, it is necessary to select five parameters, R~, K,, D¢, WA a n d WB. The first three

p a r a m e t e r s (Re, Kc, D,), are t a k e n from Refs [8-10]. In particular the q u a d r a t i c force constants Ke are calculated by n o r m a l coordinate analyses using the observed vibrational frequencies [8, 9]. The latter two ( WA, WB; electronic binding energies of free atoms), are taken from Ref. [1 I]. By m a k i n g use of these parameters, we have calculated the R-dependent electronic kinetic, electrostatic potential, a n d total molecular energies of various diatomic molecules from Eqns (14), (15), (26)-(29). Some typical examples (CH, FH) of calculated results are listed in Tables 1-6, together with the H a r t r e e - F o c k data [5]. Tables 1 a n d 2 include the numerical results of the electronic kinetic energies, Tables 3 a n d 4 those of the electrostatic potential

Table 1. Calculated and Hartree-Fock* values of the electronic kinetic energies (T)t for CH

Table 3. Calculated and Hartree-Fock* values of the electrostatic potential energies (V)t for CH

4. RESULTS AND DISCUSSION

R:~ 1.60 1.70 1.80 1.90 2.00 2.086 2.124 2.20 2.30 2.40 2.55 2.70 3.00 3.50

Calculated 38.7460 38.6171 38.5091 38.4176 38.3396 38.2812 38.2577 38.2142 38.1634 38.1188 38.0615 38.0135 37.9381 37.8529

F-H*

Error (%)§

R:~

Calculated

H-F*

Error (%)~

38.7771 38.6422 38.5266 38.4276 38.3430 38.2802 38.2550 38.2089 38.1562 38.1111 38.0556 38.0116 37.9492 37.8920

0.0802 0.0650 0.0454 0.0260 0.0089 0.0026 0.0071 0.0139 0.0189 0.0202 0.0155 0.0050 0.0293 0.1032

1.60 1.70 1.80 1.90 2.00 2.086 2.124 2.20 2.30 2.40 2.55 2.70 3.00 3.50

-76.9411 -76.8406 -76.7514 -76.6715 -76.5996 -76.5433 -76.5199 -76.4755 -76.4215 --76.3720 -76.3051 -76.2456 -76.1444 -76.0144

-76.9906 -76.8848 --76.7880 -76.7004 -76.6213 -76.5597 -76.5343 -76.4866 -76.4296 -76.3787 -76.3121 -76.2556 -76.1660 -76.0644

0.0643 0.0575 0.0477 0.0377 0.0283 0.0214 0.0188 0.0145 0.0106 0.0088 0.0092 0.0131 0.0284 0.0657

* Data taken from Ref. [5] t i n Hartree units. ~In Bohr units. §Percentage error relative to the Hartree-Fock values.

*Data taken from Ref. [5]. t i n Hartree units. :l:In Bohr units. §Percentage error relative to the Hartree-Fock values.

Table 2. Calculated and Hartree-Fock* values of the electronic kinetic energies (T)t for FH

Table 4. Calculated and Hartree-Fock* values of the electrostatic potential energies (V)t for FH

R:~ 1.325 1.37 1.427 1.50 1.58 1.65 1.696 1.7328 1.80 1.85 1.933 2.10 2.243 2.40 2.606

Calculated 100.7980 100.6960 100.5790 100.4480 100.3260 100.2330 100.1780 100.1370 100.0690 100.0250 99.9538 99.8395 99.7612 99.6908 99.6171

H-F* 100.7582 100.6430 100.5123 100.3667 100.2310 100.1296 100.0706 100.0272 99.9560 99.9089 99.8405 99.7327 99.6646 99.6087 99.5572

Error (%)~ 0.0395 0.0527 0.0664 0.0810 0.0948 0.1033 0.1073 0.1098 0.1131 0.1142 0.1135 0.1071 0.0969 0.0824 0.0602

*Data taken from Ref. [5]. t l n Hartree units. :~In Bohr units. §Percentage error relative to the Hartree-Fock values.

R~

Calculated

1.325 1.37 1.427 1.50 1.58 1.65 1.696 1.7328 1.80 1.85 1.933 2.10 2.243 2.40 2.606

-200.8470 -200.7670 -200.6730 -200.5630 -200.4540 -200.3670 -200.3140 -200.2730 -200.2040 -200.1560 --200.0810 --199.9480 - 199.8500 -- 199.7560 --199.6490

H-F* -200.7447 -200.6531 -200.5451 -200.4193 -200.2961 -200.1995 -200.1413 -200.0975 -200.0233 - 199.9725 --199.8959 --199.7663 - 199.6764 - 199.5955 - 199.5111

Error (%)§ 0.0510 0.0568 0.0638 0.0717 0.0788 0.0837 0.0863 0.0877 0.0903 0.0918 0.0926 0.0910 0.0869 0.0804 0.0691

*Data taken from Ref. [5]. t i n Hartree units. :~In Bohr units. §Percentage error relative to the Hartree-Fock values.

490

KEN OHWADA

Table 5. Calculated and Hartree-Fock* values of the total molecular energies (W)t for CH

available. F o r instance, the inhomogeneous, linear third-order differential e q u a t i o n is given by [12]

R:~

Calculated

H-F*

Error (%)~

R3(d 3 Wa/dR 3) + 9R2(d 2 Wn/dR 2)

1.60 1.70 1.80 1.90 2.00 2.086 2.124 2.20 2.30 2.40 2.55 2.70 3.00 3.50

-38.1950 -38.2235 -38.2423 -38.2539 -38.2601 -38.2621 -38.2623 -38,2613 -38.2581 -38.2533 -38.2436 -38.2321 -38.2063 -38.1614

-38.2135 -38.2426 -38.2615 -38.2727 -38.2783 -38.2796 -38.2794 -38.2777 -38.2735 -38.2676 -38.2566 -38.2440 -38.2169 -38.1723

0.0484 0.0499 0.0502 0.0491 0.0475 0.0457 0.0447 0.0428 0.0402 0.0374 0.0340 0.0311 0.0277 0.0286

+ 18R(dWn/dR)+6W n = Pn(R),

* Data taken from Ref. [5]. In Hartree units. :~ln Bohr units. §Percentage error relative to the Hartree-Fock values.

Table 6. Calculated and Hartree-Fock* values of the total molecular energies ( 140"t"for FH R:~

Calculated

1.325 1.37 1.427 1.50 1.58 1.65 1.696 1.7328 1.80 1.85 1.933 2.10 2.243 2.40 2.606

- 100.0490 - 100.0710 - 100.0940 -- 100.1140 --100.1280 - 100.1340 - 100.1360 --100.1370 - 100.1360 --100.1330 --100.1270 - 100.1080 --100.0890 - 100.0650 -100.0320

H-F* - 99.9865 - 100.0100 -- 100.0328 - 100.0526 -100.0651 --100.0670 - 100.0708 -100.0703 --100.0673 --100.0636 -100.0554 -- 100.0336 -100.0118 --99.9868 -99.9539

Error (%)~ 0.0625 0.0610 0.0612 0.0614 0.0629 0.0670 0.0652 0.0667 0.0687 0.0694 0.0716 0.0744 0.0772 0.0782 0.0781

* Data taken from Ref. [5]. t i n Hartree units. :~In Bohr units. §Percentage error relative to the Iqartree-Fock values. energies, a n d Tables 5 a n d 6 those of the total molecular energies for C H a n d FH. It is seen from these tables t h a t the calculated values agree well with the H a r t r e e - F o c k d a t a over relatively wide ranges, Therefore, we may conclude t h a t the present simple m e t h o d is adequate for predicting a p p r o x i m a t i o n s to the molecular energy c o m p o n e n t s (T~, Vn, W~) of diatomic molecules. However, to obtain more accurate predictions, further developments of this m e t h o d are required. To do this, we must construct inhomogeneous, linear higherorder differential equations [12], from which the molecular constants may be determined by using higher-order potential c o n t a n t s spectroscopically

Pn(R) = ( - l/R2)(d/dR) {R2(d/dR) [R2T,(R)]},

(30) (31)

which can be obtained by taking R times E q n (20), subsequently differentiating it with respect to R a n d finally adding the resultant to twice Eqn (20). The alternative intrinsic derivation [ 13] of E q n (30) may be performed by calculating the right-hand side terms of Eqn (32); R(d 3 Wn/dR3) + 3(d 2 W~/dR 2) -~ - d 2 ( n ] Tin )~dR 2 = - 2 [ ( n ' l Tin') + (nl T i n " ) ] ,

(32)

where I n ' ) = d l n ) / d R a n d I n " ) = d21n)/dR 2. Equation (32) is of course the second derivative of the virial theorem (Eqn (1)). By considering Eqns (30) a n d (31), we will have more accurate predictions for the molecular energy c o m p o n e n t s t h a n those o b t a i n e d by the m e t h o d described in the previous section. This will be reported in the next paper. Lastly, we add a word for the .total electrostatic potential energy V,. As is well known, V, denotes the sum of the e l e c t r o n - n u c l e a r attraction energy V,e, the electron-electron repulsion energy Vee, a n d the n u c l e a r - n u c l e a r repulsion energy Vn,. Each c o m p o nent may be obtained by introducing the homogeneity hypotheses of the total molecular a n d electronic energies with respect to the atomic n u m b e r [14, 15]. This problem also remains to be solved in the near future.

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