181
INTERNUCLEAR ENERGY FUNCTIONS FOR ALKALIHALIDEMOLECULES
Internuclear Potential Energy Functions for Alkali Halide Molecules1 by Richard L. Redington Department of Chemistry, Texas Tech University, Lubbock, Texas 79+$09 (Received May 86,1969)
Hafemeister and Zahrt have obtained values for p for twenty diatomic alkali halide molecules by fitting the equation A exp(-r/p) = KZX2/r, where 8 is the overlap of the outer shell free ion wave functions. It is shown that these p values reproduce the spectroscopic data very well when used in an overlap potential energy funcAt exp(-r/2p) ASexp(-r/3p). The coefficients, A,, correlate simply tion of the form V , = A1 exp(-r/p) with effective charge and effective distance parameters that are independent of the data. The correlation requires four parameters for the twenty molecules and returns sixty spectroscopic constants with approximately the following average errors: re (2%)) De (3%), we (6%). The overlap potential function suggests that the simple Born-Mayer expression holds best with ions of similar size and atomic number, e.g., NaF or KCl, but is poor for dissimilar ions, e.g., LiI or CsF.
+
+
Introduction An approximate internuclear potential energy function for ionic, diatomic molecules is
This function is due to Rittner,lb although it has been studied in detail by several others.2 It contains two parameters for each kind of diatomic molecule, namely, p and A , which are generally determined from experimental data. However, it is known that the repulsive potential energy used above (the Born-Mayer potential) arises in approximate quantum mechanical considerations of overlapping closed-shell systems.a In the case of the twenty alkali halide diatomic molecules, Hafemeister and Zahrt4 have listed values of p obtained by fitting the equation v4
where the S are overlaps of the outer electrons calculated using SCF free ion wavefunctions. Reliable values of A were not determined from the constant K . This equation demonstrates an accurate functional relationship between the quantum mechanical free ion overlap and r but yields only a rough approximation for the interaction potential energy due to the overlap of the ionic charge clouds. The values of p are near those reported empirically from the data for the Born-&gayer potential function; however, they are inadequate to accurately reproduce experimental results. I n the present work p is retained as a basic constant for each ion pair; however, more extensive overlap potential functions incorporating these quantum mechanical p values are appraised. Since p is independent of experimental data, this means other parameters are released for study by means of data fitting.
Vr, the Overlap Potential Function The Rittner potential function, eq 1, contains electrostatic and van der Waals attractive terms and the Born-Mayer repulsive potential. I n this section various empirical terms are added to the Born-Mayer expression in order to determine the best form for Vr, the overlap potential. The new terms turn out to be attractive or repulsive, depending upon the specific molecule. In addition to providing the basic repulsion, V zis considered to account for the perturbas tions induced by overlap on the multipole moment expansion, free ion polarizabilities, covalent tendencies, and similar concepts used to describe the system. The Hafemeister-Zahrt (H-Z) p values are used throughout. The potential energy curves are determined by forcefitting two or three of the spectroscopic constants k,, re, and De and are judged by comparing the calculated and observed ae and W e X e values. The equations for ae and W e X e are
The experimental values of (ye and W e X e are accurately known from electric resonance-molecular beam mea(1) (a) Presented at the Southwest Regional Meeting of the American Chemical Society, December 5-7, 1968, Austin, Texas. (b) E. 8. Rittner, J . Chem. Phys., 19, 1030 (1951). (2) Y. P. Varshni and R. C. Shukla, J . Mol. Spectrosc. 16, 63 (1965). (3) J. C. Slater, Phys. Rev., 32, 349 (1928); N. Rosen, ibid., 38, 255 (1931); ibid., 38, 2099 (1931); W. E. Bleick and J. E. Mayer, J . Chem. Phys., 2, 252 (1934); B. G . Dick, Jr., and A. W. Overhauser, Phys. Rev., 112, 90 (1958). (4) D . W. Hafemeister and J. D . Zahrt, J . Chem. Phys., 47, 1428 (1967). Volume 7 4 , Number 1
January 8 , 1970
RICHARD L. REDINGTON
182 Table I : Empirical Overlap Potentials, V,, Used in the Rittner Function No.
la 2b 3-5c 6-8 9-17d9e 18 19 20 21-35
Vr = F i A i e x p ( - r / p ) Fl
+ FzAzexp(--r/Zp) + F a A s e x p ( - r / 3 p ) Fa
1
exd-
PlrRitt)
Average
r
Fa
De
...
2.7% 2.0 2.3-4.0 3.3-13.4
...
2.0
1 1
*.. ... ... ... ...
...
...
.*.
..,
I . .
*..
2.1-2.3, ~ fNOS. . 3-8
...
... ...
36-38 39' 40' 41h
a,
12.6%
...
*.,
,..
I . .
...
...
--
error8
ke
7
we/Xe
7.6% 5.5 4.9-8.2 6.6-12.5 3.6-5.3 7.7 16.2 23.7 17.0-113
12.5% 5.2 7.5-13.8 10.6-24.8 6.5-8.0 5.2 14.8 15.0 5.4-92
4.5-3 * 7 5.8 6.8 16.6
7.7-7 2 8.6 10.6 36 I
a Standard Rittner function using H-Zp values; re is force-fit. Standard Rittner function from ref 2 using solid-state polarizabilities; re and k, are force-fit. Three different Vr functions. Potential 3 is: V , = Alexp( -r/p) Aaexp( - r / 2 p ) . The first of these functions is given in eq 4 and is considered to be the best overall V,. ' The force fit on no. 16 (F1 = r2, F z = r ) yields peculiar Ai values for several molecules. The average errors are a,, -28.5%; weXe, -42.4%. Expansion with orthogonal functions. This function has critical points near the experimental re values. This function force-fit the data using a constant in the exponential correction term. It yielded inferior results in addition to departing the objective of testing the H-Zp values.
+
'
surements. The values used below are taken from the tabulation of Varshni and Shukla16"with the exception of the fluoride molecule^.^^ Free-ion values for the polarizabilities6" and van der Waals constants2 are used in the calculations since the ultimate objective is a potential energy function determined completely independently of experimental data for the diatomic molecules. The results are summarized in Table I. Potential 1 is the Rittner function using the H-Z p and the
0.7 0.6
0.5 0.4
0.3
Table I1 : Potential Constants Used in Eq 4 p,
LiF LiCl LiBr LiI NaF NaCl NaBr NaI KF KC1 KBr KI RbF RbCl RbBr RbI CsF CSCl CsBr CSI a
Aa
0.252 0.299 0.311 0.335 0.264 0.305 0.316 0.340 0.299 0.333 0.344 0 365 0.314 0.346 0 357 0.377 0.336 0.364 0.375 0.394 I
I
AI X 108 ergs
0.168 0.221 0.231 0.243 0.316 0.391 0.399 0.400 0.343 0.514 0.535 0.572 0.368 0.570 0.601 0.651 0.355 0.570 0.596 0.666
Reference 4.
The Journal of Physical Chemistry
Ai X 101O ergs A i X 101" ergs
0.0557 0.523 0.947 0.118 - 0.0843 0.435 0.896 1.029 -0.484 -0.129 0.275 0,330 -0.793 -0.414 0.0558 0.0196 -1,122 -0.344 0.125 0.155
-
F
0.0704
- 0.0566
-0.182 -0.257 0.0970 0.0228 -0.136 -0.176 0.158 0.815 - 0.0196 0.0263 0.228 0.132 -0.0389 0.0257 0.303 0.0696 0.0570 -0.0563
-
-
-
CI
Br
I
Figure 1. Observed values for At using eq 4.
observed re values. This single-parameter function is not very accurate, as expected. It can be compared with potential 2, the standard Rittner function as evaluated by Varshni and Shukla.2 They determined A and p using the observed re and k, values (with solid state polarizabilities6b)and calculated De with an average error of 2Y0; the errors for ae and weXe were 5.5% and 5.270, respectively. Potential 9
V, using H-Z
p
Ale+ &e- '/'P + Ale- ' I 3 P (4) values, is forced to fit the experimental
(5) (a) Reference 2, Table 11; (b) S, E. Veazey and W. Gordy, Phys.
Rev.,138A,1303 (1965). (6) (a) L. Pauling, Proc. Roy. SOC.,A114, 181 (1927); (b) J. R. Teasman, A. H. Kahn, and W . Shockley, Phys. Rev., 92,890 (1953).
INTERNUCLEAR ENERGY FUNCTIONS FOR ALKALIHALIDEMOLECULES
183
Table 111: Spectroscopic Constantsa
A--
---re,
LiF
c1
Br
I NaF c1 Br
I KF
c1
Br I RbF c1 Br
I CsF c1
Br I
Md/A----
---ice.
kcal/m--
--De,
7-104
a, crn-1-Y
Exptl
COP
Exptl
Corb
Exptl.
Cor*
Exptl
Eq 4c
Exptl
Eq 4'
1.564 2.021 2.170 2.392 1.926 2.361 2.502 2.711 2.172 2.667 2.821 3.048 2.270 2.787 2.945 3.177 2.345 2.906 3.072 3.315
1.584 1.941 2.160 2.382 1.876 2.271 2.492 2.731 2.202 2.577 2.831 3.048 2,290 2.677 2.965 3.177 2.455 2.816 3.122 3.325
2.502 1.415 1.205 0.972 1.761 1.086 0.935 0.770 1.366 0.851 0.738 0.611 1.274 0.795 0.692 0.575 1.217 0.748 0.652 0.543
2.996 1.875 1.396 1.102 2.099 1.389 1 083 0.870 1.408 0.983 0.771 0.642 1.362 0.945 0.716 0.597 1.173 0.828 0.624 0.529
178.3 148.0 142.9 134.3 149.0 127.8 123 6 116.6 134.2 113.2 109.4 102.3 128.9 108.8 105.1 98.4 125.9 107.8 104.7 97.6
174.1 148.8 137.8 127.9 151.7 131.1 123.2 114.4 129 6 114.9 107.5 101.6 124.3 109.5 101.2 95.9 116.2 103.9 95.7 91.1
8.895 4.200 3.88 3.39 3.83 1.755 1.16 0.964 2.43 1.167 0.758 0.574 1.80 0.856 0.463 0.335 1.62 0.74 0.360 0.254
8.917 4.665 3.629 2.941 3.598 1 898 1.280 1.002 2.462 1.299 0.812 0.630 1.942 0.940 0.508 0.364 1.680 0.761 0.376 0.260
240.9 80.12 56.40 40.90 45.59 16.25 9.410 6.478 23.35 7.899 4.048 2.678 15.22 4.536 1.860 1.095 11.051 3.375 1.241 0.683
204.6 75.90 53.65 39.39 42.18 16.26 9.437 6.622 22.39 8.140 4.142 2.812 14.79 4.690 1.911 1.149 11.595 3.393 1.218 0.684
I
I
I
I
a Experimental values (rounded off) from Table 11 of ref 2, except the alkali fluorides from ref 5b. Calculated from correlated A i Ca = 3.00 X values (e0q 7); C1 = 1.00 X ca = 7.00 X 10-la; Cr = 6.00 X 10-12. The listed rs (cor) are within about rt0.005A of the true re = rmin for each potential function. Calculated using eq 4; re, k,, and D,are force-fit.
re, De, and k e values and matches the standard Rittner function in its ae and ueXe errors. None of the potential functions is significantly better than this, particularly when correlations of the three sets of twenty Ai values are attempted. This potential function is used in the following discussion; it is displayed in detail in Tables I1 and I11 and in Figures 1-3. Potential functions which depend upon quantum mechanically determined p values hold special promise.
rn 0)
0.1
0
-9 0 -0.1
1.25
1.00
-0.31,
.75
t3, 0
r9
F
50
, , , -j CI
Br
I
Figure 3. Observed values for As using eq 4.
.25 They may be valid over an extended range of internuclear distance and can, in principle, specifically include many-body interactions for polyionic species, e.g., following lines rationalized with the exchangecharge models4 As discussed in the next section, the number of empirically determined parameters can be greatly reduced if they can be accurately correlated or independently calculated, as were the H-Z p values.
0
CI -.25
-50 -.75
-1.00 F
I
I
1
CI
Br
I
Figure 2. Observed values for A2 using eq 4.
Correlation of V , for Alkali Halide Molecules The systematic variation of the Ai values seen in Figures 1-3 resembles that found for other properties Volume 74, Number 1
January 8, 1070
184
RICHARD L. REDINGTON
of these compounds. A number of correlations of the Ai values were made, with particular emphasis on relating to calculated and/or free-ion properties. The best correlation that was discovered relates the Ai values through the following parameters.
Table IV : Effective Charge and Effective Distance Parameters
LiF LiCl LiBr LiI NaF NaCl NaBr NaI KF KCl KBr KI RbF RbCl RbBr R I CsF CsCl CsBr CSI
2 . An effective charge given very simply by
N+=N+1
N-=N-1 for the cation and anion, respectively. N is the total number of electrons in the outer two shells, except for Li+ and F-, which possess only one shell. 2. An effective radius given by
R+ = ‘/z(ye
‘/z
R-
(re
- (S+ - 8-11
+ (S+ - 8-11
(6)
for the caticn and anion, where S+ and S- are the orbital maxima fcr ths outer electrons calculated by Waber and Cromer.’ These radii locate a point halfway between the ions as measured from “hard” radii, S+ and 8-, determined by the outer electron shell of each. The radii are given at the observed equilibrium internuclear distances in Table IV. Using thess parameters, the values for AI, Az, and A3 for each alkali halide are given by
- N+R+) (N+R+ - N R - ) + Cq
Az = Cz (N+R-
(7b)
Aa =
(7c)
C3
where C1, Cz, C3, and C4 are constants for the twenty molecules. These functions are plotted in Figures 4-6 against the Ai values from Table I. The experimental re values
c
N+a
s+,Ab
R+, bc
N-
S-,A
R-,A
3
0.189
9 18 25 25
0.400 0.742 0.896 1.065
11
0.278
17
0.592
27
0.734
27
0.921
0.677 0.734 0.745 0.758 0.902 0.949 0.956 0.962 1.182 1.259 1.272 1.288 I.302 1.390 1.405 1.423 1.433 1.543 1,582 1.586
0.887 1.287 1.425 1.634 1.024 1.412 1.546 1.749 0,990 1.408 1.549 1.760 0.988 1.397 1.540 1.754 0.912 1.363 1.510 1.729
OFrom eq 5. bOuter shell orbital maxima from ref 7. From eq 8, using experimental re values.
I
I
I
I
lo
I
I
1
I
I
I
1.o
2-0.5
- 1.0 -30 -20 -10
0.6 -
I
I
0
I
10 2 0 30 40
Figure 5. Correlation of observed A2 values with (N-RN+ Rt ) = A.
-
0.5 0.4 -
03
54-0.3-
0.2 A, = 1.oox
O 1 < 1
2
3
4
5
E Figure 4. Correlation of observed A1 values with (iVre)l’2, where N = N+N- / ( N + iV- ).
+
The Journal of Physical Cherniatrg
6
are used in these figures. The constants for the lines Cz = 3.00 drawn into the figures are C1 = 1.00 X X 10-l2, CS = 7.00 X 10-13, and Cq = 6.00 X 10-l2, yielding Ai in ergs/molecule, R+ and R- in angstroms, and integer values for N+ and N - . Using these constants and the re values determined from the minimum of each potential energy curve to recalculate the Ai, the spectroscopic constants for the twenty alkali halide molecules were calculated with the following average errors: re (1.7%), De (3.1%) and we (5.6%). This calculation is summarized in Table I11 and Figure 7. (7) J. T. Waber and D. T. Cromer, J. Chem. Phys., 42,4116 (1965).
185
INTERNUCLEAR ENERGY FUNCTIONS FOR ALKALIHALIDEMOLECULES
0.3 0.2 v)
L
0.1
Oa,
ro 0 F
x , Cz (3.30 X 10-l2), C3 (8.00 X 10-13), C4(-7.30 X 1O-lZ) gave re (2.5%), De (5.4%), and we (3.1%). It appears that the spectroscopic constants are returned to approximately re @%), De (3%), and we (6%).
Discussion and Conclusions The alkali halide spectroscopic properties are very sensitive to the choice of p and it is possible that a different form of potential function and/or different correlation of A i values would ensue if a different set was used. Hafemeister and Zahrt4 chose the (‘best available” free-ion SCF wave functions and found eq 2 to hold to about 0.2% in a neighborhood extending through diatomic and crystal radii. They found little contribution to p from the inner shell electrons and ignored this. It seems likely that their p values constitute a meaningful property of the interacting alkali halide ions and that they are not subject to a gross change in future calculations.* On this basis, discussion of several points concerning the overlap potential functions is justified. (1) The effective p value calculated from the quantum mechanical overlap can be used successfully in the Rittner potential when the Born-Mayer repulsive term is extended as illustrated in eq 4. Other forms tested were not as generally successful in both matching the aeand weX,values and in yielding correlatable Ai values. (2) The preexponential terms in the chosen overlap potential correlate in a simple manner with effective charge (N+,N-) and effective distance (R+,R-) par% meters (eq 5 and 6). The Az and A3 terms are observed to be smallest for the most like ions and, according to the correlation, vanish when N+R+ = N-R-. The Born-Mayer potential is seen to be poor for many alkali halides, since the potential given in eq 4 cannot be fit well with a single term, i.e., B exp(-R/b), when more than one Ai coefficient is large. The data indicate (Figures 5 and 6) the largest deviations of the BornMayer equation occur for LiI and CsF-the most dissimilar ion pairs. (3) A correlation of the Ai values with effective charge times distance parameters is anticipated by quantum mechanical approximations that express the true interaction integrals in terms of overlap integrals. The nature of the overlap integrals and their superficial resemblance to eq 7 is illustrated by S I S A , l ~ e , the simplest example (which does not occur for alkali halides). It iswritten in confocalelliptical coordinates A,p, and das 1 ( A B o A o B Z )0 ~ ’d$J * S~sA,lsu, = 7r tA3~zi-E?z(
-
where T is the internuclear separation and {A and {B are the effect’ivenuclear charges and the subscript on Ai, Bj is the exponent on hi, p i . (8) Relativistic corrections for the wave functions can reduce the overlap integrals between the heavy ions by up to 10%. Reference 4 and D. ‘CV. Hafemeister, J. Chem. Phys., 46,1929 (1967).
Volume 7 4 , Number 1 January 8,1070
RICHARD L. REDINGTON
186 Other easily obtained effective nuclear charge parameters were tried, the most successful based on the tables of Clementi and colleague^;^ however, they did not quite match the effectiveness or simplicity of N+ and N-. The effective nuclear charges were never calculated a t specific distances, e.g., R+ or R-, using SCF wave functions directly. The use of effective numbers of outer shell electrons has occurred previously in discussions of dispersion forces,1° and Pauling even brought forth their possible use for closed shell systems in his classic paper on alkali halide crystals.'l The Waber-Cromer orbital maxima are known to bear a close relationship to atomic sizes, as demonstrated by SIater,x2and (despite being alkali and halide ions) they provided the best distance parameters, giving both the best correlation and being independent of the data. The rules (eq 6) are arbitrary, but work well and yield virtually constant radii for all of the ions except I,i + and F-, as seen in Table IV. The correlation for Al seems best using eq 7a; however, nearly equivalent results are obtained for most molecules using 1
1
(a +--) N-R-
-l/a
N+R+N-RN+R+ N-R-
=( +
(9%) or
Equation 9a is identical with 7a when R+ = R-. Perhaps these formulas can be more easily derived from approximating the quantum mechanical interaction integrals than can eq 7a. It is not easy to justify the exact form of eq 7 in view of the many terms that are effectively averaged into these equations.3 McLean13 and Matcha14 have performed detailed ab initio quantum mechanical calculations of LiF13 and LiCl, NaF, NaC1, and KFI4 that reproduce the spectroscopic constants to a few per cent. The Ai values were determined for these potential curves in an attempt to determine the Ci values independently of any data. The C1 values were about 20% too high; Cz and C3 were scattered.
The Journal of Physical Chemistry
(4) At least two interesting correlation schemes for AI seem to be less effective than that given above (that is using the H-Z p values and Al only, not an effective, single term Born-Mayer potential). These are the expression derived by Hafemeister and Zahrt using the exchange-charge model4
and the Born-Mayer force equation with an equal force, f, for all moleculesl6
A~ = f
p
exp(
+ 5).
( 5 ) The four-parameter, correlated potential energy function returns reasonable spectroscopic constants, comparing well, for example, with the calculations performed by Hofer and Ferreirale on twelve alkali halide molecules. It is hoped that, ultimately, improved potential functions can be evolved, particularly those suitable for higher alkali halide molecules and for estimation of the intermolecular interactions between alkali halides molecules and rare gas atoms. The future seems bright for any effective overlap potential function between nonbonded systems (including, perhaps, core electrons in covalent bonded systems) that can be refined to the point that reliable parameters, based on computation and/or free particle properties, follow with minimal labor. Aclcnowledgments. The author is grateful t o the U. S. Army Research Office, Durham, and to the Department of Defense-Advanced Research Projects Agency, who provided financial support for this research, and to the Computer Center of Texas Tech University, who provided free computer time and facilities. (9) E. Clementi and D. L. Raimondi, J . Chem. Phys., 38,2686 (1963) ; E. Clementi, D. L. Raimondi, and W. I?. Reinhardt, ibid., 47, 1300 (1967). (10) J. Norton Wilson, ibid., 43,2564 (1965), and references therein. (11) L. Pauling, Z. Krist., 67, 377 (1928). (12) J. C. Slater, J. Chem. Phys., 41,3199 (1964). (13) A. D. McLean, ibid., 39,2653 (1963). (14) R. L. Matcha, ibid., 47, 4595 (1967); 47, 5295 (1967); 48, 335 (1968); 49, 1264 (1968). (15) M. Born and J. E. Mayer, Z. P h y s i k , 75, 1 (1932). (16) 0. C. Hofer and R. Ferreira, J . Phys. Chem., 70,85 (1966).
