1020
J . Phys. Chem. 1987, 91, 1020-1023
Theoretical Modeling of NaF: A New Configuration Interaction Potential Function and Diatomic Spectroscopic Constants P. K. Swaminathant and E. Clementi* IBM Corporation, Data Systems Division, Department 48B, Kingston, New York 12401, and National Foundation f o r Cancer Research, Bethesda. Maryland 2081 4 (Received: February 5, 1986; In Final Form: October 24, 1986)
The electronic interaction energies of NaF are obtained from computations with extensive basis set optimizations for the molecule at the configuration interaction (CI) level. The analysis of the resulting predictions for gas-phase spectroscopic properties, via a Dunham analysis, establishes that the two-body potential function deduced is of excellent quality. It also has the correct behavior for the ionic dissociation curve. There is a discrepancy between our value for the covalent dissociation energy and the experimentally reported wide range of such values for NaF, which are larger than the value obtained at present by about 15-24 kcal/mol. This discrepancy may arise almost entirely from the well-known difficulty of theoreticallyreproducing the electron affinity of fluorine.
I. Introduction Recently, we started’J a detailed investigation of the electronic interaction energies of alkali metal halides with the aim of studying the binding forces in their gaseous phase as well as their condensed-phase manifestations, namely molten salts and crystals. Two papers from this group dealt with NaCI; the present article summarizes new results for the two-body electronic interaction energies of NaF. As in the case of other alkali metal halides, all the calculations3 to date have treated the electronic problem of NaF, at the Hartree-Fock level of approximation, with only moderate success for the prediction of gas-phase behavior. Whereas such results3 are interesting, we found striking improvements in the gas-phase spectroscopic properties when we included electron correlation effects using a configuration interaction (CI) study for the case of NaC1.’ In this paper, we present similar improvements in the quantum mechanical results for N a F in the gas phase. Via Dunham4 analysis, we calculate the predictions of the present potential to be in excellent agreement with experimental results5 for vibrational and rotational spectroscopic constants and Dunham coefficients for this molecule. In order to extract useful ab initio models of condensed-phase properties, the present interaction potential must be suitably combined with future three-body, four-body, etc. quantum mechanical interaction energy computations. Such a systematic procedure can in principle yield quantitative estimates of microscopic many-body effects on the macroscopic properties of condensed phases. The prerequisite of studying three-body effects in the many-body expansion of the interaction energy is an accurate two-body-level description and the progress reported here for the ionic interaction in N a F fulfills this. The configuration interaction study is described in some detail in section 11. The gas-phase spectroscopic properties are presented and analyzed in section 111. Section IV contains some concluding remarks. 11. Basis Set and the Electronic Interaction Energies
Following the strategy that we used’ successfully for NaCl of optimizing the basis sets distance by distance, we aimed to represent well the ionic N a + F in the equilibrium region while at the same time smoothly going over to the covalent N a F when the atoms are further out. Again, we optimized the molecule itself, thus avoiding reliance on the atomics, and used configuration interaction results to locate minima with respect to the scaling parameter C of the geometric basis set.6 The framework of the “geometrical basis set”6 uses C, the scaling parameter, common ‘Present address: Suite 106, 9560 Pennsylvania Avenue, Upper Marlboro, MD 20772. *Author to whom correspondence should be addressed.
0022-3654/87/2091-lO20$01.50/0
to all the Gaussian basis functions, and a geometric progression of exponents that are suitably nested in going from function types s to p and d, etc. so as to provide a convenient handle on basis set optimizations as well as save computer time in the evaluation of integrals. For each internuclear separation, the molecular calculation optimized with respect to C by using a minimum of three C values per distance. The size of all the Gaussian basis sets used for fluorine and sodium were of size (18,12,2)(10,6,2). The initial optimizations (exponents as well as contractions) were done for the atoms Na and F and the ions Na+ and F at the SCF level, for each given value of C. Two d functions without contractions were then added to the resultant basis sets for atomic Na and ionic F-; the latter species were chosen in every case, because, with the F(’S) basis set, we could readily reproduce F(*P) and, equivalently with Na@), we could reproduce well the energy of Na+(’S). To generate the CI, we used S C F level wave functions from the program package PP-1-1982.’ Our CI calculations used the MRDCI program of Buenker,* which provides a multireference CI and uses a configuration selection procedure9 based on perturbation theory combined with an energy threshold criterion. The details of this method as well as extensive applications of it to ground and excited electronic state calculations have been discussedand reviewed beforeS9As before,’ we performed both an open-shell S C F and a closed-shell S C F for a given internuclear distance, thus providing alternative starting wave functions for the CI program. The open-shell S C F was useful to represent the covalent N a F singlet state at large distances. Although in the case of NaCl this procedure determined an appropriate point of domination by the ionic or the covalent state, that problem is not thus resolved in NaF. All the MRD-CI were performed using 12 most significant reference configurations, chosen by the criterion of maximum energy lowering. Naturally, this included the covalent N a F and the ionic N,+F-. The CI matrix was generated by keeping the 1s core orbitals frozen. An energy threshold of 50 microhartrees for configuration selection was used resulting (1) Swaminathan, P. K.; Laaksonen, A,; Corongiu, G.; Clementi, E.
Theoretical Modeling of NaCI in the Gas, Solid and Liquid Phases: Possible Role of Many-Body Forces; IBM Research Report KGN-6, 1984. J. Chem. Phys. 1986, 84, 867. (2) Laaksonen, A,; Corongiu, G.; Clementi, E. Int. J . Quanrum Chem., Quantum Chem. Symp. 1984, S18, 131. (3) Matcha, R. L. J . Chem. Phys. 1968,48,335, for NaCI; J . Chem. Phys. 1967, 47, 4595, for LiCI; J. Chem. Phys. 1967, 47, 5295, for NaF. (4) Dunham, J. L. Phys. Rev. 1932, 41, 721. ( 5 ) Veazey, S . E.; Gordy, W. Phys. Rev. A 1965, 138, 1303. (6) Clementi, E.; Corongiu, G. Chem. Phys. Lett. 1982, 90, 359; and for tables see: IBM Technical Report POK-11, 1982. (7) IBMOL: Ortoleva, E.; Castiglione, G.; Clementi, E. Comput. Phys. Commun. 1980, 19, 337. (8) Buenker, R . , private communication. (9) Buenker, R. J.; Peyerimhoff, S . D. In Exited Stares in Quantum Chemistry; Nicolaides, C. A,, Beck, D. R., Eds.; Reidel: Dordecht, Holland,
1979.
0 1987 American Chemical Society
Theoretical Modeling of N a F in C I matrix dimensions in the range 2000-10,000. The positions of the exponents (with the magnitudes of the exponents result from the scheme of the geometric basis set's6 defining process itself) for the d-functions were optimized for the molecule, near equilibrium geometry, at the C I level, by using two reference configurations, N a F and Na'F. These positions of the two d-functions were kept fixed, for all values of C , while adding polarization functions before closed-shell molecular computations. Although we tried adding in one f-function on each of the atomic centers, the calculated covalent dissociation energy for N a F was only negligibly different from the value without the f-functions. Therefore, the present potential functions were generated from the above basis set, without adding on the ffunctions. A singular measure of the extent of our success lies in the accuracy of the covalent dissociation energy, requiring a good description of both covalent and ionic states. Our results for the covalent dissociation energy of NaF, 99.2 kcal/mol, obtained from the optimized basis, is almost 14.8-23.8 kcal/mol lower compared to the range where experiments lie (considering the thermochemical limit,'ObJOc1 14 kcal/mol, and the recommended value, 123 kcal/mol, of ref loa). When a Davidson correction" is used to obtain a full CI estimate of 103.6 kcal/mol the discrepancy remains at 10.4-19.4 kcal/mol. The difficulty of achieving a sufficiently accurate representation of the atomic and ionic species of fluorine, namely the fluorine electron affinity problem,I2 is the primary cause of our poor covalent dissociation energies. We can say this with confidence, because the present basis set for sodium is even better than what we had in the NaCl calculations,' where the ionization potential of Na was already well represented, as demonstrated by the accurate results obtained there. It is also knownI2that a significant part of the fluorine electron correlation comes from triple and quadruple excitations. Although MRDCI does include the effects of some of the higher-than-doubles excitations via the multireference feature, the present example of fluorine is a rather subtle one, always requiring special attention. In fact, this problem remains, to date, incompletely characterized in the literature. The representation of both states is primarily important in the coupling region between them where they exhibit avoided crossing and maximum competition to dominate. One cannot locate the crossing point very accurately unless both the states are represented very well. In spite of the above discrepancy, the present excellent basis set can describe well the ionic interaction energies for N a F and so we have obtained these. But we did not attempt to obtain a covalent singlet curve for N a F in this work. As in the NaCl case, we observed a nonconvergence in the closed-shell molecular computations, as we approached the pseudocrossing region, making direct determination of the Na'F dissociation limit unreliable. So, as before, we determine the ionic dissociation limit using the expected long-range pure Coulombic behavior of the counterionic dissociation curve. The procedure we adopted to obtain the ionic interaction energy potential function from our C I results used the criterion that the behavior be Coulombic for internuclear distances beyond r > 1 1.5 bohrs. This particular choice of r for NaF, is not arbitrary, but based on choosing the point that gives the correct ionic dissociation energy. The region of strong competition between the covalent and the ionic states appears before r < 11.5 bohrs for NaF. In this respect, there is a qualitative difference from NaC1, where a pure Coulombic behavior was appropriately placed beginning at shorter internuclear distances than was the coupling region. The different procedure used above for N a F was adopted because the Coulombic curve beyond r k 11.5 bohrs could be smoothly connected to the (10) (a) Ham, D. 0. J . Chem. Phys. 1974,60, 1802. (b) Gaydon, A. G. Dissociation Energies, Chapman and Hall: London, 1968. (c) Brewer, L.; Brackett, E. Chem. Rev. 1961, 61, 425. (1 1) Davidson, E. R. In The World of Quantum Chemistry; Daudel, R., h l l m a n n , B., Eds.; Reidel: Dordrecht, Holland, 1974; p 17. (12) Sasaki, F.; Yoshimine, M. Phys. Rev. A 3974, 9, 26. Barnett, R. N.; Reynolds, P. J.; Lester, Jr., W. A. J . Chem. Phys. 1986, 84, 4992 and references therein.
The Journal of Physical Chemistry, Vol. 91, No. 5, 1987 1021 TABLE I: Collected Extrapolated CI Energies for N a + F (Long-Range Coulomb Tail beyond r > 11 bohrs) Fixing the Reference Point as the Ionic N a + F Singlet State interaction distance, energy, point bohrs kcal/mol -6.296 -34.674 -66.206 -90.264 -108.531 -122.079 -124.294 -127.456 -132.08 1 -144.330 -147.580 -150.624 -150.630 -1 50.090 -149.030 -147.618 -144.135 -140.100 -135.746 -126.772 -1 16.274 -98.698 -85.652 -67.982 -60.062 -54.566 -50.200 -46.482
2.50 2.60 2.70 2.80 2.90 3.00 3.02 3.05 3.10 3.20 3.40 3.65 3.70 3.80 3.90 4.00 4.20 4.40 4.60 5.00 5.50 6.50 7.50 9.50 10.50 11.50 12.50 13.50
7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
NaF Ionic State Curves 200
-200
-.-
I
1
Rittner ab initio B-H-M-F-T
I
I
1
I
4
7
10
13
16
R (bohrs)
Figure 1. Various N a + F (Le., ionic) interaction laws. Shown along with the present results are the BHMFT curve'* and the Rittner potential."
value of the potential at r = 7.5 bohrs by using the fitting functional form adopted here (see below), and it avoided the use of the possibly inaccurate values for the potential obtained in the strongly coupled region. The ionic interaction energies are collected in Table I, and the potential curve is shown in Figure 1. The coupling region (7.5 < r < 11 bohrs) in Figure 1 is smoothly linked by the fitted form. The fitted result is visually indistinguishable from the ab initio curve on the scale of the figure. Also shown in Figure 1 is the comparison to popularly used ionic potentials for NaF. It is seen that our potential is deeper than the empirical potentials near equilibrium. In this region, it is closer to the RittnerI3 potential than to the crystal-based BHMFTI4 (Le., the Born-HugginsMayer-Fumi-Tosi parametrization) potential function. (13) Rittner, E. S. J . Chem. Phys. 1951, 19, 1030. (14) Fumi, F. G.; Tosi, M. P. J . Chem. Solids 1964, 25, 31. Tosi, M. P.; Fumi, F. G. J . Chem. Solids 1964, 25, 45.
1022
Swaminathan and Clementi
The Journal of Physical Chemistry, Vol. 91, No. 5, I987
TABLE III: Total Energies, Bond Lengths, and Dissociation Energies for NaF from This and Other Works
TABLE 11: Extrawlated MRDCI Energies for F X F
Doint 1 2
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
F'
distance, bohrs
interaction energy, kcal/mol
2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00 7.00 8.00 9.00 10.00 11.oo 2.00 3.00 4.00 5.00
266.871 21 3.941 174.753 146.811 129.787 117.732 108.364 100.495 87.775 77.616 69.327 62.68 1 57.561 52.428 48.864 44.992 42.111
total
energy, au -261.84700 -261.87442 -261.3785 -621.4385
.F' interact ion Curve
200
-
n
a >
3.665
99.2 103.6 70.3 123 (+3)* 114'
3.628 3.640"
remarks this work, CI this work (Davidson") ref 3
exptl
based on ab initio potentials. One will have to include that state also in a calculation with a properly adjusted basis set that treats the crossing correctly. We also simply note here that the F?- curve obtained here does not even result in a bound solid lattice when combined with the NaF curve obtained here. In fact this was what got us initially involved with the question of what might be wrong with the F-F potential obtained here.
ab initio
Y 0 v
dissocn energy, kcal/mol
Reference 5 . Reference 1Oa. Reference 1Ob,c.
250
'1
bond length, bohrs
150
111. Potential Energy Functions and Spectroscopic Constants Some of the characteristic properties of our quantum mechanical potential, the equilibrium energy, bond length, and dissociation energy are listed in Table 111. The table compares our values to results from previous theoretical works as well as an experimental work. Our bond length agrees well with the experimental one. The dissociation energies were discussed in the last section. We use an analytical expression for the ionic potential energy function that was used in our previous works:'%2
V(r) = F o u L ( r )
100
v
+ PR(r)
where 50
FouL(r) = Z,ZJe2/4acor
(2)
+
(3)
+
V,SR(r)= (Al,e-BrJr C,r)e-D?ir ElJfll~ 0 1
4
7
10
R (bohrs
13
16
)
Figure 2. The present F X F interaction potential surface of Table 11.
We also calculated the F-F repulsion curve using the present basis set for fluoride, and the results for this curve are given in Table I1 and Figure 2. The F-F- curve was calculated from S C F and CI (with one reference state) to be only slightly different. However, this potential curve is to be regarded as an exploratory calculation. Our preliminary potential energy curve obtained already shows indication that it will cross with the autoionized state F2- e- as seen from an estimate based on our results for F22- and those of Lauderdale, McCurdy and HaziI5 for F2-. Asymptotically, these curves will be separated by the exact electron affinity of flurine,I2 say 3.4 eV, with the F22-curve below that of F2- in this region. The F2- curve will be flat compared to the Coulomb curve of F2- as the internuclear separation is reduced. The F22-curve becomes more repulsive than the Coulomb curve between 7-10 bohrs and we estimate that it should cross the F2curve between 4-7 bohrs. Of course, the present F22- curve is probably in gross error due to this crossing. Since F; is an open-shell case, it was possible in ref 15 to find the point of breakdown of the S C F (corresponding to autoionization) by watching for a change to positive sign of the orbital energy associated with the last lone electron. In our closed-shell case of F,2-, it is not possible to thus locate such a breakdown of S C F since the highest orbital will contain one electron even after autoionization and hence the orbital energy will not change sign even if autoionization occurs. The fact that a crossing with the autoionized species mcurs for the F22-diatomic curve is a significant finding for modelling studies
+
(15) Lauderdale, J. G.; McCurdy, C. W.; Hazi, A. V . J . Chem. Phys. 1983. 7 9 , 2200.
FouL(r)is the electrostatic interaction between the ions, whereas p R ( r )describes the short-range part of the interactions. The same form is used for both like-ionic as well as counterionic potential fits. The potential parameters AII-F,Jof eq 1-3 are given in Table IV, which also contains the corresponding parameters for the potentials between the like cations calculated in a previous work.2 The maximum error in the nonlinear fit of the Na+-F- pair potential is less than 0.88 kcal/mol for all the points and 0.30 kcal/mol for the points in the binding region. The error in the fit for the like ions is less than 0.1 kcal/mol for all the points. The fitting parameters quoted in Table IV do not have any physical significance but are pure fitting parameters. Calculations of spectroscopic constants are based on the series expansion method by D ~ n h a m .The ~ potential energy function is expanded in the powers of ( R- R,)/R, and the time-independent Schrodinger equation is solved for the vibrating rotor. The solutions for the stationary states are then related to the spectral constants YIJin the energy term expression (4) The calculated vibrational constants are given in Table V together with the experimental values of Veazey and G ~ r d y Also . ~ included in Table V are the theoretical values of M a t ~ h a .Our ~ calculated value for the fundamental vibration frequency w,(Y,,) is 533.0 cm-' and is 3.1 cm-' away from the experimental microwave value, 536.10 f 0.4 cm-I, reported by Veazey and G ~ r d y .The ~ improvement from Matcha's3 reported value of 558.4 cm-' can be attributed to (1) the carefully optimized geometrical basis set and (2) the effect of the CI. Whereas we have two d-functions, Matcha includes one d and one f polarization functions in his most accurate basis set. Thus the primitive Gaussian basis functions that were used by Matcha are of comparable quality to a single set of the present basis functions. The CI level treatment of electron cor-
The Journal ofPhysica1 Chemistry, Vol. 91, No. 5, 1987
Theoretical Modeling of N a F TABLE I V Petential Parameters of Eq 1-3' A B -0.301 70 140.383 99 0.12897
-0.940 10 -4.611 12 -0.929 02
C
D
E
F
interaction
-3.337 22 -0.257 01 -0.087 46
1.33603 1.181 66 0.103 15
25.424 97 13.190 09 2.553 50
-3.750 71 5.511 62 -1.31901
+++ --
1023
"All values are in atomic units.
TABLE V Calculated Values (in em-') of Vibrational Constants for NaF
4YIO) W e X A Y20)
wae(y30) wcze(~40)
x
lo1 103
Matcha"
exptb
558.3 4.387 0.6954 2.503
536.1 3.830
this work 533.0 3.562 0.1315 1.364
"Reference 3. bReference 5.
a0 a1
a2 a3 a4 aS
Matchab
exptC
present
177400 -3.257 6.602
164450 -3.133 1 6.433 -9.200
163500 -3.092 6.483 -11.24 17.57 -25.82 36.43
a6
TABLE VI: Calculated Values (in em-') of Rotational Constants for NaF Matcha" 4.393 4.679 3.184 10.88 39.11 6.922
TABLE VII: Calculated Values (in em-') of Potential Coefficients for the Dunham Expansion of Potential Energy"
exptb 4.369 4.557 2.282 11.61 7.005 4.105 1.156 3.736
this work 4.343 4.442 1.557 -11.53 34.89 -2.821 1.476 -0.002 -2.786 0.3495 0.4460
"Reference 3. bReference 5.
relation effects is also responsible for the improvement obtained in the slopes of the potential. Because the basis set that is optimum for C I is not the one that is optimum for SCF, we omit detailed comparisons in terms of our SCF energies with the S C F results of Matcha. Our theoretical values for the anharmonicity constant w,xe of 3.562 cm-' compares very favorably to the experimental value of 3.83 cm-l. There are no experimental values reported for w a e and wezein the literature. Our theoretical rotational constants are given in Table VI. The present calculated value for the rotational constant Be, 0.4343 cm-', is very close to the reported microwave5 value, 0.4369 cm-'. The other rotational constants show different degrees of agreement. The agreement between theoretical and experimental values for ae,0.4557 X (0.4443 X and De, 0.1161 X lV5(0.1153 X (all values are in cm-I; the experimental values are in parentheses), is excellent, while agreement is less satisfactory for -ye, 0.2282 X 10-4 (0.1557 X l p ) , Be, 0.349 X lo4 (0.701 X lo-"), H,0.28 X lo-* (0.41 X lo4), and He, 0.28 X (0.41 X 10-l2). We also report calculations of Y,, Yo4, Y13,YZ2,and Y3' from Dunham analysis. However, these constants have not been observed experimentally. Calculated Dunham potential coefficients are given in Table VII. Our theoretical coefficients are in good agreement with the ~ the experimental a,,, ..., a3 values of Veazey and G ~ r d y .For
' a , denotes the coefficient o f f where x = (R- R,)/R,. bReference 3. 'Reference 5.
higher order potential coefficients, there are no experimental results for comparison.
IV. Concluding Remarks The present potential energy function of the ionic singlet state of N a F is found to be an accurate gas-phase potential in spite of the difficulty arising from the fluorine electron affinity problem that leads to a less-than-perfect covalent dissociation energy. It is not difficult to find explanations for the larger discrepancy in dissociation energy observed here when compared to our previous study of NaC1. The order of magnitude of the discrepancy indicates the origins to be the triple and higher excitations coming into the fluorine electron affinity problem12 and not completely built into the MRDC19 level of C I calculation adopted here. In addition, there may be a small contribution arising from incompleteness of the basis set, largest for the fluorine electron affinity physics where up to i-functions have been used before.I2 The ionic curve presented here is accurate enough to be combined with future calculations of three-, four-, etc.-body interaction energies to build a many-body interaction expansion representation, to be used to model condensed-phase properties. Such expansions have been tested and interesting microscopic models explored successfully for the example of waterI6 from this group. However, the case of ionic solids and liquids lies at another extreme where electron correlation effects that appear at the two-body level seemI7 to be compensated at some higher body level and thus constitute a more challenging example to unravel. Acknowledgment. P.K.S. thanks the National Foundation for Cancer Research for support for part of the research conducted pursuant to a contract with the foundation. Also, P.K.S. is grateful to Prof. R. Buenker for helpful suggestions with regard to the use of the MRDCI program and to Prof. C. W. McCurdy for conversations on the autoionization phenomenon. Registry No. NaF, 7681-49-4; F2, 7782-41-4. (16) Clementi, E.; Corongiu, G. Int. J . Quantum Chem. 1983.10, 31; also IBM Research Report POK-20,1983, and Detrich, J.; Corongiu, G.; Clementi, E. Chem. Phys. Lett. 1984, 112, 426. (17) Fowler, P. W.; Madden, P. A. Mol. Phys. 1983,49, 913.
