J. Phys. Chem. 1983, 87,
detailed studies of the spin- and direction-dependent configuration-interaction coupling strengths. Finally, it is interesting to point out that, when these extra degrees of freedom are included, the problem becomes identical with that of molecular fluorescence in a collision context,13 in which the photon frequency w , wave vector direction R, and polarization direction ii corr_espondto E , k (electronic momentum unit vector), and S (electronic (13) K.-S. Lam and T.F. George, J. Chem. Phys., 76, 3396 (1982).
2803-2810
2803
spin unit vector) in the problem of collisional ionization.
Acknowledgment. This work was supported in part by the National Science Foundation under Grant CHE8022874 and the Air Force Office of Scientific Research (AFSC), United States Air Force, under Grant AFOSR82-0046. The US. Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation hereon. T.F.G. acknowledges the Camille and Henry Dreyfus Foundation for a Teacher-Scholar Award (1975-84).
A Self-consistent Field Interaction Energy Decomposition Study of 12 Hydrogen-Bonded Dimers W. A. Sokalski,+ P. C. Harlharan, and Joyce J. Kaufman' DepeHment of Chemistry, The Johns Hopkins University, Baltimore, Matyland 2 12 18 (Received: October 25, 1982)
The recently introduced SCF interaction energy AESCF decomposition scheme corrected for basis set superposition error (BSSE) has been employed in a systematic study of 12 hydrogen-bonded dimers formed by CH,, NH3, H20,and HF. The calculations have been performed in STO-3G, 4-31G, UQ-A, and MODPOT basis sets and compared with some 6-31G* and 6-31G** results. The observed basis set dependence of A&CF originates mainly from the first-order electrostatic contribution and in particular from the monopole-monopole term in the atomic multipole expansion. The surprisingly good results obtained in the minimal all valence effective core model potential (MODPOT)and uniform quality (UQ-A) basis sets are due to proper monomer charge distribution in these basis sets. In addition a large BSSE has been found for the weakest complexes in 6-31G* and 6-31G** basis sets. STO-3G results, even when corrected for BSSE, seem to be completely unsatisfactory.
1. Introduction Ab initio LCAO MO SCF method has been widely used in systematic studies on the nature of hydrogen bonding' where the SCF interaction energy AEsCF was partitioned into the following contributions:2 AEscF + BSSE = E D - EA - EB = EL + EX + PL + CT + MIX (1)
2. Method Theoretical interaction energies may be related to the experimentally determined equilibrium dimerization constants K,, standard free energy AGTO, enthalpy AHTO, or entropy AST" changes -RT In K , = AGT" = A€€," - TAST" = AE + A ( E - Uoo)- ART In ( q / N ) (2)
EA, EB,and ED denote the total energy of A, B, and AB
where A(E - U,") is the zero-point vibrational energy change, q , the statistical sum, and N , the number of particles. Unfortunately the nonempirical estimates6 of thermodynamic functions display strong basis set dependence originating mainly from AESCF contribution to AE
systems; EL represents electrostatic, EX, exchange, PL, polarization, CT, charge transfer, and MIX, the coupling term. Unfortunately some of these interaction energy contributions contain a nonphysical basis set superposition error (BSSE).3 Therefore the first aim of this study is to reinvestigate the nature of contributions to AEscF in the framework of the recently improved AEscF decomposition4 where the BSSE has been removed by means of the counterpoise method (CP).3 The second aim is to interpret the observed basis set dependence of AEsCp This extends our earlier investigations5 leading to the concept of an efficient basis set for evaluation of first-order interaction energies. 'Visiting scientist. Permanent address: Institute of Organic and Physical Chemistry 1-4, Wyspiaiiskiego 27,50-370 Wroclaw, Poland. 0022-3654/83/2O07-2803%0 7.5010
(1) H. Umeyama and K. Morokuma, J. Am. Chem. SOC.,99, 1316 (1977). (2) H. Kitaura and K. Morokuma, Int. J. Quantum Chem., 10, 325 (1976). (3) S. F. Boys and F. Bernardi, Mol. Phys., 19, 558 (1970). (4) W. A. Sokalski, S.Roszak, P. C. Hariharan, and Joyce J. Kaufman, Int. J . Quantum Chem., Congress Issue, in press. (5) W. A. Sokalski, P. C. Hariharan, and Joyce J. Kaufman, J.Comp. Chem., in press. (6) P. Hobza, P. Carsky, and R. Zahradnik, Int. J.Quantum Chem., 16, 257 (1979).
0 1983 American Chemical Society
2004
Sokalski et al.
The Journal of Physical Chemistry, Vol. 87, No. 15, 1983
where E#SPdenotes the second-order dispersion contribution. The standard SCF supermolecular approach includes basis set superposition error.3 In addition, even AEscF corrected for BSSE are still basis set dependent.718 To analyze the source of the observed dependence our modified AEsCF decomposition scheme4has been employed
[kcal A6/mol]. Equation 8 takes into account the anisotropy of dispersion interactions and higher cg, Cl0, etc., terms in molecular expansion. However, atom-atom CF parameters cannot be directly compared with experimental C, values. The dispersion term from ref 13 leads to significant improvement of the simulated liquid water properties15and has also been confirmed in experimental studies.16
where E,(,, and E,(,, denote the total energies of isolated A and B molecules obtained in the AB supermolecule basis set, EgL is the first-order electrostatic term identical with the EL term from eq 1,E& is the first-order exchange free of first-order BSSE"), and is the second-order induction term. Ei3Dhas been further decomposed into the classical long-range contribution E!#D,LEreferring to locally excited configurations (LE) only-identical with PL in eq 1, and short-range charge transfer (CT) contribution Ei#D,CT free of BSSE-in contrast to CT terms in eq 1
3. Geometries and Basis Sets All monomer units were held a t experimental geometries, i.e., HF: r(FH) = 0.917 A;17 HzO: r(OH) = 0.957 A, LHOH = 104.5;18 NH,: r(NH) = 1.012 A, LHNH = 106.7°;19CH4: r(CN) = 1.094 A, LHCH = 109.47.20 The geometry of each linear hydrogen-bonded dimer H,X-. HYH, is defined by distance r(XY) and angle 0 spanned between the X-HY hydrogen-bond axis and the line connecting electron donor X center of gravity of HEX. As the aim of this work is to compare various basis sets, the same dimer geometry has been used in our calculations for different basis sets (Table I). The structure of the nine dimers formed by NH,, HzO, and HF has been taken primarily from the studyz1(uncorrected for BSSE) using the 6-31G* basis setz2or from experiment. For H,N--HCH3 and Hz0-.HCH3 complexes the geometry' optimized within 4-31G basis setz3(uncorrected for BSSE) has been adopted. Additionally, the structure of the HF-HCH, system has been optimized in this work in the same manner. To relate our results to some previous s t u d i e ~we l ~have ~~ evaluated all interaction energy components in the minimal STO-3G set,25in the split valence 4-31G23basis sets, in the minimal P4P basis set of uniform quality,26and in the minimal all valence 3s3p MODPOT basis setz7where inner-shell electrons are represented by ab-initio effective core model potentials. From our recent study5 the last two basis sets appear to be well-suited for proper representation of the first-order interaction energy. Finally, the corresponding AEscF interaction energies corrected for BSSE have been evaluated in extended 6-31G*22and 631G**28basis sets with polarization functions and compared with STO-3G, 4-31G, UQ-A, and MODPOT results. Our results had been compared also with the corresponding values obtained in minimal OLDz3and NEW7 basis sets optimized by Kolos to preserve the proper description of intermolecular interaction energies.
To analyze the most important electrostatic interaction energy & !E in more detail its multipole contribution E&MTP has been separated from the remaining penetration term
E& = EfiL,,Mw+ E~L,PEN
(6)
where the multipole term has been approximated by the sum of the atomic monopole-monopole atomic monopole-dipole E&q,pand atomic dipole-dipole EELt,, contributions E#L,MTP =
E&,qq + E&,qr + E&+,
(7)
For comparison we also evaluate the molecular dipoledipole interaction energy E&DD. The SCF interaction energy AEscF defined above has to be supplemented by the dispersion term E&,p. Evaluation of its exact value by means of CI, MC-SCF, or even by a perturbation approach may make the whole calculation much more costly than AEscF itself since extended basis sets with polarization functions are essential.'O E&P values obtained within any exact method in minimal basis set will be substantially in error.g Use of the Unsold approximation has been shown to lead to reasonable values of dispersion energies even when minimal basis sets are used." While there are fewer virtual orbitals with a minimal basis set, the average excitation energy is approximately the same as with larger basis sets.12 In the case of large molecular systems when the use of extended basis sets is expensive we prefer to estimate the dispersion term from atom-atom C6 coefficients, fitting accurate dispersion energies for azabenzene dimers," (HzO)z,13 and
@WZl4 E&p
-c
asA bfB
(CpCbb)1/zRab-6
(8)
where Rabis the interatomic distance and CfH = 26.98, C'! = 649.6, CTN = 503.94,l' C ': = 155.95,13and CzF = 60.9814 (7) W. Kolos, Theor. Chim. Acta, 54, 187 (1980). (8) F. P. Groen and F. B. Duijneveldt, Theor. Chim. Acta, in press. (9) W. A. Sokalski and H. Chojnacki, Int. J. Quantum Chem., 13,679 (1978). (10) F. Mulder, G.van Dijk, and C. Huiszoon, Mol. Phys., 38, 577 (1979). (11) C. Huiszoon and F. Mulder, Mol. Phys., 38, 1497 (1979); errata, 40, 249 (1980). (12) A. van der Avoird, personal discussion at the 4th International Congress of Quantum Chemistry, Uppsala, Sweden, June 1982. (13) B. Jeziorski and M. van Hemert, Mol. Phys., 31, 713 (1976). (14) H. Lischka, J . Am. Chem. SOC.,96, 4761 (1974).
4. Results and Discussion The corrected SCF stabilization energies
AEsCF,
basis
G. C. Lie and E. Clementi, J . Chem. Phys., 62, 2195 (1975). G. D. Zeiss and W. J. Meath, Mol. Phys., 29, 161 (1975). G. Herzberg, "The Spectra of Diatomic Molecules", Van Nostrand, Princeton, NJ, 1950. (18) W. S.Benedict, N. Gailar, and E. K. Plyler, J. Chem. Phys., 24, 1139 (1956). (19) K. Kuchitsu, J. P. Guillory, and L. S. Bartell, J. Chem. Phys., 49, 2488 (1968). -.__ ~
(20) G. Herzberg, "Electronic Spectra of Polyatomic Molecules", Van Nostrand, New York, 1966. (21) J. D. Dill, L. C. Allen, W. C. Topp, .~and J. A. Pople, J. Am. Chem. SOC.,97, 7220 (1975). (22) P. C. Hariharan and J. A. Pople, Mol. Phys., 27, 209 (1974). (23) R. Ditchfield, W. J. Hehre. and J. A. Poule, J . Chem. Phvs., 51, 2657 (1969). (24) W. A. Lathan, L. A. Curtiss, W. J. Hehre, J. B. Lisle, and J. A. Pople, Prog. Phys. Org. Chem., 11, 175 (1974). (25) W. J. Hehre, R. F. Stewart, and J. A. Pople, J. Chem. Phys., 31, 2657 (1969). (26) P. G. Mezey and I. G. Csizmadia, Can. J. Chem., 55, 1181 (1977). (27) H. E. Popkie and Joyce J. Kaufman, Int. J . Quantum Chem., 10, 47 (1976). (28) P. C. Hariharan and J. A. Pople, Theor. C h m . Acta, 28, 213 (1973).
The Journal of Physical Chemistv, Vol. 87, No. 15, 1983 2805
SCF Study of Hydrogen-Bonded Dimers
A:
D:
HF W H P H, t-F Hp N Y NH, NH, CH4 CH4 CH4 NH, H;D NH, Hp HF I-tF NH, y0 HF NH, H$ HF
A :
D:
HF HF H O , HF H$ NH, NH, NH, CH, CH, CH, \IH, y0 N Y H-D HF HF N Y I $H HF NH, H O , HF
0 r!
f
1
2
Y
-,2
w" a -4
I
I I
I I
,n
I I I I
-STO-3G ---1-31G
-6
UQ-A
- -6-31G - -6-31G.m -MODPOT
-0
-IC
J
Flgure 2. SCF interaction energies A€sF corrected for BSSE for 12 hydrogen-bonded dimers in 6 different basis sets. 1'
Flgure 1. Basis set superposition errors (BSSE) for 12 hydrogenbonded dimers in 6 various basis sets (A, electron acceptor; 0,electron donor molecule).
set superposition errors (BSSE), and dispersion energy estimates E&+ calculated as described in section 2, eq 8, for all 12 dimers considered, are given in Table I and compared with some previously published values in minimal OLD,29NEW,7 and Gaussian Hartree-Fock A 0 (HFA0)30basis sets and, where available, with AI3 estimates from experimental s t ~ d i e s . ~ l -The ~ ~ observed differences between the best theoretical and experimental estimates may be attributed mainly to the lack of intramolecular correlation effects and improper representation of the dispersion term. Additionally, those geometries optimized in the 6-31G* basis set include BSFIE.~~ The equilibrium hydrogen bond lengths RXH-Ygiven in the first column of Table I in parentheses have been determined in this work in 4-31G basis set (corrected for BSSE). Both minimal 7s3P OLD29and NEW' basis sets have recently been derived to reproduce properly interaction energies while minimizing BSSE. To illustrate the size of BSSE in different basis seta the corresponding values have been plotted in Figure 1where all 1 2 complexes are ordered according to the decreasing W. Kolos, Theor. Chim. Acta, 51,219 (1979).
P.A. Kollman and L. C. Allen, J. Chem. Phys., 52,5085(1970). (31)J. W.Bevan, A. C. Legan, D. W. Milles, and S. C. Rogers, J.
Chem. SOC.,Chem. Commun., 130,341 (1975). (32)D. G. Lister and P. Palmieri, J. Mol. Struct., 39, 295 (1977). (33)R. W.Bolander, J. L. Kassner, and J. T. Zung, J. Chem. Phys., 50., 4402 (1969). ~~.~ (34)P.A. Kollman in 'Methods of Electronic Structure Theory", H. F. Schaefer, Ed., Plenum, New York, 1977,p 109. (35)H. A. Gebbie, W. J. Borroughs, J. Chamberlain,J. E. Harris, and R. G. Jones, Nature (London),221, 143 (1969). (36)D.F. Smith, J. Mol. Spectrosc., 3, 473 (1959). (37)E. V. Frank and F. Meyer, 2.Elektrochem., 63, 577 (1959). (38)J. S.Rowlinson, Trans; Faraday SOC.,45,974 (1949). (39)J. E. Lowder, J. Quant. Spectrosc. Radiat. Transfer, 10, 1085 (1970). ~~
interaction energy. The same ordering has been preserved in all the figures. Unexpectedly, BSSE in 6-31G* and 6-31G** basis sets for the five weakest complexes is quite significant and much larger than those obtained in minimal basis sets especially in minimal all valence MODPOT basis set. Therefore uncorrected 6-31G* or 6-31G** results may differ considerably from the corresponding Hartree-Fock limit. In all cases the share of BSSE in total interaction energy remains nonnegligible. The basis set dependence of corrected SCF interaction energies aEsCF is presented in Figure 2. The largest deviations approach -10 kcal/mol. Generally, 4-31G results seem to overestimate and STO-3G to underestimate the accurate interaction energies represented here by corrected 6-31G* and 6-31G** values. In addition, the corrected STO-3G AEscF energies frequently lead to qualitatively incorrect predictions of the relative stability of molecular complexes. Considerably better predictions may be obtained in minimal 783POLD,29 NEW,7 and uniform quality 8s4P26basis sets except for complexes involving methane. However, the best AEsCF values have been obtained with the modest (but previously carefully optimizedz7)minimal all valence 3s3p MODPOT basis set.27 The observed basis set dependence has been analyzed within the AESCF decomposition scheme corrected for BSSE4. In Figure 3, the first-order E(l) contributions presented have been separated from the higher order terms aEsCF - E(') (Figure 4). Apparently, the observed basis set dependence of A E ~ c Foriginates from E(1).The slight basis set dependence of higher order terms - E(') N E&, seems to be due to basis set size. Therefore one may expect a slight increases of these contributions with increase of basis set size. The close agreement between 4-31G and (11,7,2/6,1) [4,3,2/2,1] extended basis set induction energies for water dimer (-1.70 and -1.63 kcal/mol, respectively) in-
-
The Journal of Physical Chemistry, Vol. 87, No. 15, 1983
2806
Sokalski et ai.
TABLE I : Comparison of Interaction Energies A E ~ C AE, F , and BSSE in (kcallmol) in Various Basis Sets
MODPOT 6-31G*
basis set
STO-3G
4-31G
UQ-A
AESCF
-5.72 -4.74 -10.46
-14.66 -1.41 -16.07
H,N...HF: r ( N F ) = 2.77 A (2.44 A ). -9.87 -11.44 ' -11.29 ' -11.13 -1.89 -0.92 -0.89 -0.89 -12.36 -11.75 -12.18 -12.02
-3.28 -9.00
-3.28 -17.94
-3.28 -13.15
-1.97 -5.98 -7.95
-11.74 -1.42 -13.16
H,O.,,HF: r(OF) = -8.38 -8.82 -2.49 -1.43 -10.87 -10.26
Et;tSP
-2.43 -4.40
-2.43 -14.17
-2.43 -10.81
AESCF
AESCF+
-2.65 -2.89 - 5.54
AE
_____-
BSSE
AESCF+
6-31G**
OLD0
NEWb
HFAOC
-11.88 -2.35 -14.23
-10.05 - 2.00 -12.05
-11.7
-1.94 -13.82
-1.94 -11.99
2.68 A (2.687 A ) , e = 0" -8.28 -8.02 -9.03 -0.94 -0.16 -3.52 -9.22 -8.18 -12.55
-7.47 -2.48 -9.95
e
expt
= 0"
BSSE
EHSP AE AESCF
BSSE
i.3 E SCF'
-3.28 -14.72
-3.28 -14.57
-3.28 -14.41
-9.40
BSSE
-1.33 -10.36
-1.33 -8.80
-7.75 -1.02 -8.77
H,N.,.HOH: r ( N 0 ) = 3.05 A (3.00 A ) , 0 = 0" -5.13 -5.76 -5.77 -5.68 -6.02 -1.37 -0.94 -0.71 -0.66 -1.95 -6.48 -6.34 -7.97 -6.50 -6.70
-5.09 -1.57 -6.66
-2.16 -4.81
-2.16 -9.91
-2.16 -7.29
-1.13 -7.15
-1.33 -6.22
-1.69 -3.42 -5.11
-6.22 -1.44 -7.66
H,O...HOH: r ( O O ) = 2.98 A ( 2 . 9 6 A ) ; e = 0" -4.15 -4.43 -4.70 -4.57 -4.88 -2.13 -0.87 -1.43 -0.63 -2.58 -6.28 -5.86 -5.57 -5.20 -7.46
-4.4 5 -1.61 -6.06
Et;fSP
-1.54 -3.23
-1.54 -7.76
-1.54 -5.69
-0.75 -5.63
-0.75 -5.20
AESCF
0.27 -4.63 -4.36
-4.80 -2.12 -6.92
HF..,HF: r(FF) = -3.11 -3.48 -2.76 -2.01 -5.87 -5.49
2.79 A ( 2 . 8 1 A ) , 8 = 72" -4.18 -4.16 -3.81 -1.55 -1.62 -4.12 -5.73 -5.78 -7.93
-2.91 -2.40 -5.31
E6'tSP
-1.24 -0.97
-1.24 -6.04
-1.24 -4.35
AESCF AESCF+
-0.74 -1.90 -2.64
-3.41 -1.74 -5.15
HF...HOH: r ( F 0 ) = -2.18 -2.36 -1.63 -1.06 -3.81 -3.42
%rsP AE
-0.82 -1.56
-0.82 -4.23
-0.82 -3.00
-0.43 -2.89
-0.43 -2.63
AESCF
-1.54 -1.19 - 2.73
-3.23 -0.81 -4.04
H,N...HNH,: r ( N N ) = 3.44 .A ( 3 . 3 4 A ) , e = 0" -2.23 -2.46 -2.41 -2.41 -2.65 -0.78 -0.63 -0.52 -0.53 -1.16 -3.01 -3.09 -2.93 -2.94 -3.81
-2.29 -0.77 -3.06
-1.18 -3.90
-1.18 -4.41
-1.18 -3.41
-1.23 -0.98 -2.21
-2.66 -1.22 --3.88
-0.75 -3.98
-0.75 -5.41
-0.75 -4.65
-1.68 -1.69 -3.37
HF.,.HNH,: r ( F N ) = -1.15 -1.32 -1.07 -0.56 -2.22 -1.88
3.38 14 ( 3 . 4 1 A ) , e = 82.8"
-0.63 -0.85 -1.48
-1.25 -1.39 -2.64
--1.21 -1.46 -2.67
-1.42 -1.35 -2.77
-1.28 -0.46 -1.74
-0.52 -1.15
-0.52 -2.20
-0.52 -1.67
-0.52 -1.77
-0.52 -1.73
-0.37 -1.79
-0.37 -1.65
AESCFt
-0.21 -0.23 -0.44
-0.37 -0.73 -0.10
!-I,N...HCH,: r ( C N ) = 4.02 .A (4.105 A ) , B = 0" +O.l5 -0.47 -0.25 -0.27 -0.28 -0.19 -0.48 -0.51 -0.13 -0.66 -0.73 -0.78
YSP
-0.48 -0.69
-0.48 -0.85
-0.48 -0.33
AE
BSSE
-2.43 -11.25
-2.43 -10.71
-2.43 -10.45
-9.90d
-5.80
BSSE
AESCF
BSSE
AESCFt
-2.16 -7.92
-2.16 -7.93
-2.16 -7.84
-5.30
BSSE
AE
BSSE
4ESCFt
-1.54 -5.97
-1.54 -6.24
-1.54 -6.11
-4.60e -6. 90f
-4.60
BSSE
AE
BSSE
-1.24 -4.72
-1.24 -5.42
-1.24 -5.40
-0.86 -4.67
-0.86 -3.77
3.10 .A (3.04 A ) , e = 69.8" -2.62 -2.61 -2.46 -1.38 -1.41 -2.24 -4.00 -4.02 -4.70
-2.20 -1.03 -3.23
-5.40g -7.40h
-3.00
BSSE
BSSE
AESCFf
-0.82 -3.18
-0.82 -3.44
-0.82 -3.43
-2.70
BSSE E&P
AE
AESCF
BSSE
AESCF+
-1.18 -3.64
H,O.,,HNH,: r ( 0 N ) = -1.90 -2.05 -1.05 -0.63 -2.68 -2.95
-1.18 -3.59
-1.18 -3.59
-0.67 -3.32
-0.67 -2.96
3.43 A (3.31 A ) , e = 68.7" -2.00 -1.96 -2.21 -0.85 -0.93 -1.49 -2.85 -2.89 -3.70
-2.00 -0.71 -2.71
-0.75 -2.75
-0.54 -2.54
-4.70' -5.5d
-2.30
BSSE
W S P
AE
AESCF
BSSE
AESCFi
-0.75 -4.80
-0.75 -2.71
BSSE E%P
AE
AESCF
BSSE
BSSE E(2) A
-0.52 -1.84
-0.48 -0.95
-0.48 -0.73
-0.48 -0.75
-0.54 -2.75
-1.30
The Journal of Physical Chemistry, Vol. 87, No. 15, 1983 2807
SCF Study of Hydrogen-Bonded Dimers
TABLE I (Continued)
UQ-A MODPOT 6-31G*
STO-3G
4-31G
AESCF
-0.07 -0.33 -0.40
-0.14 -0.91 -1.05
t0.30 -0.48
Egisp
-0.42 -0.49
-0.42 -0.56
AESCF
-0.06 -0.04 -0.10
-0.12 -0.45 -0.57
-0.42 -0.42 -0.42 -0.42 -0.68 -0.12 -0.48 -0.50 HF...HCH,: r(CF) = 4 . 2 A (4.175 A ) , e = 0" +0.05 -0.16 -0.09 -0.09 -0.09 -0.03 -0.42 -0.44 -0.04 -0.19 -0.51 -0.53
-0.12 -0.18
-0.12 -0.24
BSSD
AESCF+
BSSE
AE
BSSE AESCF*
6-31G**
OLD"
basisset
NEWb
HFAOC
expt
H,O...HCH,: r ( C 0 ) = 3.8 A , e = 0" -0.26 -0.06 -0.08 -0.25 -0.69 -0.77 -0.18 -0.51 -0.75 -0.85
BSSE %lSP
AE
a Reference 29. Reference 7. and 35. g References 34 and 36.
A:
D
:
-0.12 -0.12 -0.12 -0.12 -0.07 -0.28 -0.21 -0.21 Reference 30. Reference 31 and 32. e References 33 and 34. References 34 References 34 and 37., * References 34 and 38. References 34 and 39.
HF HF H$ H$ HF H-p NH, NH, NH, CH,, CH, CH4 NH, H20 N Y Hp t-F HF N 5 H.p HF NH, HJI M
A : HF HF HzO HP HF HP NH, NH, NH, CHL CHL CH4 D : NH, H,0 NH, HP HF HF NH, H20 HF NH, HzO HF
n /
f7 o
I
\
12 \
Y
\
-
u 1
2
,-,
fl'
>
. . . ...
: 4
LI
;k
-2 / / / - - - - ' '/ / - /
4
/
I
/
,-I
-6
I!'" i
,
-a
0 -% T n- 'r, US"
.lL
-4
- - L-31G UQ -A
-e
-MODPOT -12
i
J
- 16
/-I
-20 -10
Flgure 3. First-order interaction energies E(') for 12 hydrogen-bonded dimers in 4 different basis sets. HF HF H P HP HF HP NH, NH, NH, CH, CH, CH, NH, HP NH, HF HF NH, Hj2 HF NH, H P HF
HP
STO -3G I-'
/
/
/ i
+I
- --
4-31G
....
uQ -A
-MODPOT
Figure 4. Higher order contrlbutiOns MScF - E(')to the SCF interaction energy corrected for BSSE for 12 dimers in 4 basis sets.
Figure 5. First-order electrostatic 622 and exchange 6Li contributions for 12 dimers in 4 basis sets.
dicates that already the split valence basis set may be sufficient to ensure saturated induction term E@,. Also the MODPOT, UQ-A, and STO-3G values of EfND (-1.29, -1.37, and -1.42 kcal/mol, respectively) are quite reasonable. Next, the first-order interaction energy has been decomposed into its electrostatic E#. and exchange E& constituents (Figure 5). The basis set dependence of E(') is mostly due to the electrostatic term. The minor dependence of the exchange E& constituent may be ascribed to the inadequate long-range behavior of the wave funct i ~ n The . ~ electrostatic term E!& has been further split into its multipole Egl,MTP(Figure 6c) and penetration EfiL,PEN (Figure 6a) contributions where the first atomic monopole-monopole term E & , (Figure 6b) from the atomic multipole contribution seems to be responsible for the basis set dependence. So the quality of the resulting AEscF may be partly related to proper charge distribution. Similar conclusions have been drawn already in our preliminary ~ t u d i e s .This ~ explains also the success
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2808
The Journal of Physical Chemistry, Voi. 87, No. 15, 1983
Sokalski et al.
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The Journal of Physical Chemlstty, Vol. 87, No. 15, 1983 2809
SCF Study of Hydrogen-Bonded Dimers
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of the uniform quality UQ-A and 383PMODPOT basis sets derived to reproduce the charge distribution in extended basis sets. Finally, instead of using atomic multipole expansion we show in Figure 7 the first term in the molecular expansion E!&DD. The molecular dipole-dipole term E&DDis frequently used as a crude estimate of the total interaction energy. Obviously this term is not representative as the relative stabilities of complexes are not reproduced correctly (Figure 7). So the molecular dipole moments cannot be used to derive scaling factors correcting the basis set dependence of interaction energies.' All previously discussed interaction energy components are given in Table I1 with the corresponding basis set superposition errors decomposed into first-order BSSE'') and higher order BSSE(R)contributions4 BSSE = BSSE'')
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2810
J. Phys. Chem. 1983, 87, 2810-2815
(d) The quality of charge distribution cannot be judged solely on the basis of the molecular dipole moments as the molecular dipole-dipole term may frequently lead to 5. Conclusions qualitatively incorrect predictions of relative stabilities of (a) In contrast to previous studies' the relative impordifferent molecular complexes. tance of various interaction energy components in linear (e) Extended basis set 6-31G* and 6-3G** results for hydrogen-bonded complexes may be described as weak molecular complexes contain nonnegligible BSSE. (0 STO-3G SCF interaction energies even corrected for BSSE yield qualitatively wrong stability predictions for strong molecular complexes. The relative orientation of interacting monomers is almost (g) Higher order induction interaction may be reproentirely determined by the electrostatic term.4 duced correctly already within split valence basis sets (b) The basis set dependence of the counterpoise corprovided the BSSE has been separated from the secondrected SCF interaction energy originates mainly from the point charge EkLt,, term from the multipole part E&MTP order contribution. Even the minimal all-electron and all-valence MODPOT basis sets ensure the reasonable of the first-order electrostatic Ek/- interaction energy. of Et#,. Similar conclusions may be also found e l ~ e w h e r e . ~ ~ ~representation ~~~~~' (c) As the contribution of the inner-shell electrons is Acknowledgment. This work has been supported in part negligible4,5(and, moreover, with customary all electron by NCI under Contract N01-CP-75929, in part by ONR basis sets including the inner shell, part of the BSSE is Power Programs Branch under Contract N00014-80-Cdue to the inadequate representations of the inner shells), 30003, and in part by the Polish Academy of Sciences one may obtain reasonable results within minimal all vawithin the MR 1-9 Project. W.A.S. is grateful to Professor lence basis sets correctly reproducing the molecular charge W. Koios and Drs. B. Jeziorski and G. Chalasihski for distribution. reading the preliminary version of the of the manuscript and helpful discussion. Registry No. Methane, 74-82-8; ammonia, 7664-41-7; water, (40) G . Karlstrom and A. J. Sadlej, Theor. Chim. Acta, 61, 1 (1982). 7732-18-5; hydrogen fluoride, 7664-39-3. (41) W. A. Sokalski, J . Chem. Phys., 77, 4529 (1982). of BSSE has been discussed extensively in our earlier paper.*
Some Remarks on the Density Functional Theory of Few-Electron Systems Robert G. Parr" and Llbero J. Bartolotti Department of Chemistry, University of North Carolina, Chapel Hi//, North Carolina 27514 (Received: December 3, 1982)
The density functional theory for the ground state of an electronic system is re-formulated by replacing the electron density p by the product of the number of electrons Nand a shape factor u, and several features of the theory are elaborated in this formulation. Clarification is thereby provided for a number of subtle points in the theory for a finite number of electrons, especially aspects of the chemical potential p. Among the questions considered are definition of the energy curve E(N) and possible discontinuities in its slope p , ambiguities in various functional derivatives and long-range behavior of them, and similar questions in Kohn-Sham theory. It is shown that without knowledge of an appropriate general functional E[p],including its dependence on N, it is impossible to determine its chemical potential, from either the actual electron density of a system or a calculation of any kind on that system holding N constant. Nevertheless the chemical potential is well defined and physically meaningful for any specific system of interest.
I. Preface This paper is intended to provide, among other things, complete discussion of a quite basic question: Given an exact electron density for the ground state of some specific atom or molecule of interest, how, if a t all, can one determine, in principle and (hopefully) in practice, the electronic chemical potential of the system? The chemical potential here is the Lagrange multiplier p in the central stationary principle of density functional theory 0022-3054/83/2087-28 10$01.50/0
d{E[p,u]- p ( N [ p ] - N)) = 0
for constant u
(1)
where E [ p , u ] is the energy as a functional of the electron density p and
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N is the number of electrons of the system of interest, which has the Hamiltonian
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P
0 1983 American Chemical Society
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