CLASSICAL THEORY OF ALKALI HALIDE MOLECULES

BY YATENDRA PAL VARSHNI Dept. of Physics, Allahabad University, Allahabad, India

Received 28th May, 1956

Calculations of rotational constant xe and vibrational constant wge for alkali halide molecules have been carried out assuming two types of repulsion terms in the potential energy function, viz. inverse power and exponential. By comparing the calculated values of Cte and o d e with the observed ones it has been concluded that the exponential type of repulsion term is satisfactory.

The theory of ionic lattices and the concept of electrostatic valence forces (Born and Goeppert Mayer 1) has proved to be very successful in the treatment of polar crystals, especially alkali halide crystals. Its success is directly connected with the circumstance that for the atomic distances in the neighbourhood of the minimum of the potential curve, the electronic charge distribution is such that it can be approached by the concept of the purely ionic bond. According to this approximation the energy is divided in two parts :

(a) attractive, consisting of a purely ionic term and polarization terms, etc.; (b) a short-range repulsive term. In the crystalline state of many substances known as polar compounds, the

ionic concept has proved to be in quantitative agreement with the experiment within experimental error.

Several attempts have been made to give a classical concept of alkali halide gas molecules, analogous to Born’s lattice theory. The earlier attempts were due to Born and Heisenberg,2 Fajans and Schwartz,3 Verwey and de Boer,4.5 May 6 and Rice.7

Most of these calculations were based on the assumption that the repulsive constants are the same for the gas molecules as for the crystals, an assumption which was later found to be untenable. Further no experimental data for inter- nuclear distances in gaseous state were available. Due to these reasons the earlier calculations did not yield good results.

Later, Verwey and de Boer 5 employed directly the internuclear distances determined by electron diffraction and were able to achieve a high measure of success. Recently, Rittner 8 has rejected the use of crystal data and employed the internuclear distance in the molecule, along with force constant, obtained from molecular spectra, and he has b=en able to account successfully for the im- portant properties of the molecule (see also Trischka 9).

More recently Honig, Mandel, Stitch and Townes 10 have compared the co- efficients of higher powers of (r - re) in the expansion of energy expression in terms of (r - re) as obtained from experiment with those from Rittner’s theory. They found satisfactory agreement considering the large uncertainties in some of the experimental data.

Attempts haw been made to apply Rittner’s theory to gaseous alkali hydrides (Klemperer and Margrave,ll Altshuller 12) and thallous halides (Altshuller 13).

It was found by Brewer and Mastick 14 that binding in the alkali halides may be fairly well described by using only e2/re for the ionic binding energy. Similar results were found by Margrave,lS for the gaseous diatomic hydrides and halides

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Y. P. VARSHNI 133

of group 2 and 3 metals. It seems that in the gaseous state, distortion of the positive and negative ions may occur to a great extent, thereby leading to an increase in the binding energy (polarization energy), which is almost exactly counterbalanced by the repulsive energy.

In the classical theoretical treatments of ionic molecules both types of repulsive terms have been used. The original Born inverse power relation,

B(r) = b/rn, (1)

(2) Though for the crystalline state the exponential form has been found to be

better than the inverse power term (Mayer and Helmholtz,l6 Huggins 1 9 , for gaseous diatoms it is still an open question.

The wave function in the outer sphere of the ions falls off exponentially, and an exponential repulsion law may, therefore, be expected for small overlappings of the wave functions. In the diatomic molecules, however, the overlapping is considerably larger, since the internuclear distance is smaller ; the wave functions are much more distorted from their original spherical symmetry. Investigations of Verwey and de Boer 49 5 showed that repulsion increases more rapidly with decreasing r than given by the exponential function. Using the inverse power repulsion term, they found excellent values for the binding energies.

In view of the fact that in recent years very accurate data of molecular constants of alkali halides have become available, it seemed worthwhile to investigate the relative merits of the two types of repulsion terms. In the present paper we will compare the theoretically calculated values of the rotational constant and vibrational constant W,X, for the two types of repulsion terms with the experimental values.

as well as the Born-Mayer exponential law, B(r) = A exp (- rip).

THEORY

The details of the theory have been given by Rittner.8 Here we give only an outline.

The alkali halide molecule is considered to be constituted of ions, each of which is polarized by the electrostatic field of the other.

Let + e, - e be the charges on ions, X I , a2 the polarizabilities, and PI, p2 the induced moments. The major part of the binding energy arises from the electrostatic interactions between the ions : charge-charge interaction, charge- dipole interaction, dipole-dipole interaction, and quasi-elastic energy stored in the induced dipoles. Following Born and Heisenberg,2 the portion 4 of the total binding energy U may be expressed as a function of the internuclear distance r as follows :

It can be shown (Rittner,8 Debye 18) that

ex1 2ex1~2 r2 r5 p1=-+- + . . .,

Substituting (4) and ( 5 ) in (3)

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134 A L K A L I H A L I D E MOLECULES

The remainder of the binding energy is treated "in the spirit of Born and Mayer's 19 lattice theory " ; thus we have further a repulsion term, a van der Waals attractive term, and kinetic energy term representing the difference in translational, rotational, and vibrational energy between the molecules and free ions of which it is composed. The total energy is then

u= --- e2 e2(aI + a2) - ~ - - - 2e2ala2 c hvo hV0 f-+ r 2r4 r7 r6 2 exp (hvo/kT) - 1

both types of the repulsion term have been given. Applying the conditions,

(dU/dr), = 0, (d2U/dr2), = ke, we obtain

2e2 lOe2(ccl + a2) + 112e2ala2 42c kere + - + + -

+- + - 2 re + re5 reg re7

2e2 10e2(al + a2) 112e2ala2 42c k e = - - - -____-- re3 re6 re9 reg

re7 - re e2 2e2(al + a2) 14e2ala2 6c P '

-- re2 re5 reg n + l =

A closed analytical potential energy function can be expanded as follows : 1 1 1

2! 3! 4 ! U(r) = - Uxl(re)(r - re)2 + - UII*(re)(r - re)3 + - UIv(re)(r - re)4 + . . . .

Let U1"(re)/U1'(re) = X, and U*V(re)/UII(re) = Y.

By solving the wave equation for (12), Dunham 20 has shown that

a, =- (q +

WeXe = -x2- Y - [: ]s&- If we take the reduced mass in atomic weight units ( P A ) then

2.1078 x 10-16 P A

wexe = [; x2 - Y ]

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Y. P. V A R S H N I 135

For the inverse power energy expression, completely new calculations have been made. For the exponential energy expression, certain calculations of Honig, Mandel, Stitch, and Townes have been utilized. Our X and Y, in terms of their d 1 and d 2 are

X = 3 d l / r , ; Y = 1 2 d 2 / r e 2 .

Hence

Van der Waals’ c can be calculated from (Pauling and Wilson 21),

3 I 2 E

2 I2 + E’ c = - cc1cc2 ____

where 12 is the second ionization potential of the alkali atom, and E is the electron affinity of the halogen atom.

Tables 1 and 2 give the necessary data. Data given in table 1 have been taken from Rittner’s 8 paper. Table 3 shows the computed value of c and n.

Li Na K Rb c s F c1 Br I

LiI NaCl NaBr NaI K F KCI KBr KI RbF RbCl RbBr RbI CsF CSCl CsBr CSI

TABLE 1

a 2 (10-24 cm3)

1 *05 3.69 4.8 1 7-16

TABLE 2

ke (10s dyneslcrn)

~7845 1.186 1 -044

1 -20s 1.019 *818 -704

1.361 1 -076 *788 -633

1 -45 1 -95 -83 -55 1

-9378

I (10-12 ergs)

120.6 75.30 50.79 43.74 37.5

re (10-8 cm)

2.392 H2 2.361 ,, 2.502 ,, 2.711 ,, 2.55 G4 2.667 H2 2.821 ,, 3.048 ,, 2.246 e H3 2.787 H2 2.945 H2 3.177 ,, 2.345 ,, 2.906 ,, 3.072 ,, 3.515 ,,

E (10-12 ergs)

6.618 5.960 5.59 1 5.030

Be

-4429 H2 -2179 ,, el511 ,, -1177 ,, a2022 ,, -1285 ,, -0811 ,, -0608 ,,

-0876 ,, -0475 ,, ~0328 ,, -184 ,, .072 ,, -036 ,, .0236 ,,

e, estimated value ; B1, Barrow and Caunt ; 23 H2, Honig, Mandel, Stitch and Townes ; 17 H3, same as H2 ; calculated by semi-empirical formula.

Calculated values of a, and wexe, by the two expressions along with the ob- served values, have been given in tables 4 and 5 respectively. Except where other- wise indicated, data have been taken from Herzberg.22

G4, Grabner and Hughes ; 24

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136 A L K A L I H A L I D E M O L E C U L E S

TABLE 3

diatom

LiI NaCl NaBr NaI KF KCl KBr KI RbF RbCl RbBr RbI CsF CSCl CsBr CSI

LiI NaCl NaBr NaI KF KC1 KBr KI RbF RbCl RbBr RbI CsF CSCl CsBr CSI

diatom

LiI NaCl NaBr NaI KF KCl KBr KI RbF RbCl RbBr RbI CsF CSCl CsBr CSI

obs.

W409 H2 -00161 ,, -00094 ,) 40065 ,, -

-00079 ,, WO405 ), -00027 ,,

40045 9 ,

-000186 ,, .o0011 ,, .00110 ,, -00033 ,) *o0012 ,, -000068 ,,

-

C (10-60)

1.50 5-46 6.06 9.17 7.74 24.8 30-5 41.3 12.8 42.2 50-8 68.8 21.6 69.4 85.5 116.2

TABLE 4

n

4.828 6.391 6.503 6.940 8-5 17 7.549 7.225 7.53 1 6.591 8.543 7.708 7.654 7-222 8.403 8.603 7.582

TABLE 5

a e calc. (inverse power)

-00552 -0020 -001 18 400847 40203 .00102 -000508 *000344 .00200 -000623 400237 -000 1 40 -00159 -000446 -000163 40008 14

Wexe d c . (inverse power)

3.47 2.37 1.71 1.48 3.06 1 *74 1 -03

2-43 1 -62

*837

-664 -462 2-40 1-16 *602 -258

(exponential) a, calc.

-00196 -00151 -0o0909 400674

400812 400403 400277

400498 *000190 .0o0113 -00125 400367 -000140 *oooO705

-

-

Wexe C ~ C . (exponential)

-91 1.83 1.35 1-20

1.36 -802 -676

1-20 -528 -380

-946 -526 -272

1-94

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Y. P . V A R S H N I 137

DISCUSSION

It will be observed from table 4 that except for LiI, the calculated values of ae using the “ exponential term ” are in good agreement with the experimental values. Results using the “ inverse power term ” are poor ; the calculated values are about 20 % higher than the observed ones.

Also the calculated values of wexe using the “exponential term” are much better than those using the “inverse power term”. But even the exponential values, except for LiI, are always greater than the experimental values. However, it should be remembered that the wex, values given by Barrow and Caunt 23 are not strictly experimental but calculated from the empirical formula,

Since Rittner’s theory well reproduces D, and cc,, it appears that the experi- mental values of WeXe require to be increased.

Results for LiI are bad, both for a, and w,x,. For this molecule re and CCe

are known with great accuracy. Thus we are left with possible errors in we.

This supports the view put forward in a previous paper (Varshni and Majumdar 25) that its me is incorrect. We are led to conclude that the exponential term is satis- factory for the gaseous diatoms of alkali halides, and the inverse power term is not.

The author is thankful to Prof. K. Banerjee for his kind interest in the work, and to Mr. S. N. Srivastava for assistance in the calculations and preparation of the manuscript. Thanks are also due to Dr. J. W. Trischka (Syracuse) and Dr. A. Honig (Paris) for making available their data on alkali halides prior to publication.

1 Born and Goeppert-Mayer, Handbuch der Physik, 1933, 24 (2), 623. 2 Born and Heisenberg, 2. Physik, 1924, 23, 388. 3 Fajans and Schwartz, Z. physik. Chem., Bodestein Festband, 1931, 717. 4 Verwey and de Boer, Rec. trav. chim., 1936, 55, 431. 5 Verwey and de Boer, Rec. trav. chim., 1940, 59, 633. 6 May, Physic. Rev., 1937, 52, 339 ; 1938, 54, 629. 7 Rice, Electronic Structure and Chemical Binding (McGraw Hill Book Co., Inc.,

8 Rittner, J . Chem. Physics, 1951, 19, 1030. 9 Trischka, J. Chem. Physics, 1952, 20, 181 1. 10 Honig, Mandel, Stitch and Townes, Physic. Rev., 1954, 96, 629. 11 Klemperer and Margrave, J. Chem. Physics, 1952, 20, 527. 12 Altshuller, J . Chem. Physics, 1953, 21, 2074-75. 13 Altshuller, J . Chem. Physics, 1953, 21, 2074. 14 Brewer and Mastick, J. Amer. Chem. Soc., 1951, 73, 2045. 15 Margrave, J . Physic. Chem., 1954, 58, 258. 16 Mayer and Helmholtz, 2. Physik, 1932, 75, 19. 17 Huggins, J . Chem. Physics, 1937, 5, 143. 18 Debye, Polar Molecules (Dover Publications, New York, 1945), p. 60. 19 Born and Mayer, 2. Physik, 1932, 75, 1. 20 Dunham, Physic. Rev., 1932, 41, 713, 721. 21 Pauling and Wilson, Introduction to Quantum Mechanics (McGraw Hill Book Co.

22 Herzberg, Spectra of Diatomic Molecules (D. van Nostrand Co., Inc., New York,

23 Barrow and Caunt, Proc. Roy. Soc. A , 1953, 219, 120. 24 Grabner and Hughes, Physic. Rev., 1950, 79, 819. 25 Varshni and Majumdar, Indian J. Physics, 1955, 29, 38.

New York, 1940), p. 249.

Inc., New York, 1935).

1950).

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