Localized orbitals in hydrogen bonded systems

Volume 27, number

CHEMICAL

2

LOCALIZED

ORBITALS

PHYSICS

LETTERS

IN HYDROGEN

15 July 1974

BONDED

SYSTEMS

P. LINDNER* and John R. SABIN Quantum Theory Project and Departments of Physics and Chemistry, Gainesville, Florida 3261 I, USA

Received

22 April 1974

The localized molecular orbitals for a variety of hydrogen bonded CND0/2 wavefunctions. The bonding of these systems is qualitatively

Since the realization that hydrogen bonds play an important role in the determination of the secondary structure and function of many large, biologically important organic molecules, it has become increasingly important to understand the various contributions to hydrogen bonding. The molecules with which one has to deal become increasingly large as the biological realm is entered, and quantum mechanical calculations become consequently more costly and cumbersome. It would be desirable, then, to examine some of the common features of hydrogen bonds, with a view to obtaining transferable molecular fragments, which could be used to describe the hydrogen bonds in large systems without carrying out a full calculation. A good deal of progress has already been made, both in the description of properties of hydrogen bonds [l-3] and in the notion of transferability [4] . The classical chemical valence picture of hydrogen bonds is most useful in this regard. Here the bonds are described in terms of lone pairs and localized bonds between atoms, which are ideally suited for transference between similar species. Although it has been well known for some time [S] that such a simple description does not adequately describe hydrogen bonds, there is a quantum mechanical analogue of this localized description, namely the “localized mob

Uppsala

214

address: University,

systems are obtained from their INDO and discussed in light of the localized orbitals.

lecular orbitals” (LMO’s) of Edmiston

1. Introduction

* Permanent

University of Florida,

Department of Quantum Chemistry, S-751 20 Uppsala 1, Sweden.

and Ruedenberg

(ER) l&71. As the LMO method has already been shown to give a good description of non-classical systems like B2H6 [8,9] , which contains bridging hydrogens, it might also be expected to give reasonable results on hydrogen bonded systems. We have thus carried out a variety of semiempirical calculations coupled with the Edmiston-Ruedenberg localization scheme on a number of simple hydrogen bonded systems. The aim of this work is twofold. It is intended to test the suitability of localization, in the INDO and CND0/2 models [lo] (for a description of these methods, see ref. [ 113 ), for describing the hydrogen bonded systems. Secondly we hope to gain some qualitative understanding of this type of bonding in a localized molecular orbital picture.

2. Method of calculation The electron density for a molecular given by

system is

(1) if the wavefunction is approximated by a single determinant constructed from a set of occupied one-electron orbitals, {pi(r)}, e.g., the canonical HartreeFock orbitals [ 121. Any unitary transformation of the originally chosen orbitals leaves the electron density

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unchanged. This invariance property can be used to visualize the electron density as a superposition of local densities corresponding to bonds, lone-pairs and core-electrons. The localization procedure of Edmiston and Ruedenberg [6,7] provides such a description, in terms of localized molecular orbitals. These orbitals, are chosen to maximize the total self-energy, D, of the electrons:

(2) They are derived from a unitary transformation on SCF determined molecular orbitals. The Edmiston-Ruedenberg localization scheme has to be treated with care if it is coupled with a semi-empirical method, which heavily relies on a zerodifferential overlap approximation [lo] . Especially the CNDO/2 method can give rise to indeterminate results for systems containing conjugated parts or involving atoms with two lone-pairs [8] . On the other hand, the less approximate INDO method seems to be more reliable in mimicking the “ab initio” results [8]. In the present note we are interested in the qualitative features of a “localized hydrogen bond”. We therefore feel justified in using the INDO model coupled with the Edmiston-Ruedenberg (ER) localization scheme. We also feel that the present results are of general interest as most f the larger molecules which interact through hydrogen bonds at present usually are described in a CNDO/2 or INDO model. For the B2H, molecule we have compared our semi-empirical results with some “ab initio” calculations. As we, in the present work, are interested in how “localized hydrogen bonds” look in some semiempirical methods in common use, we did not pursue the “ab initio” comparison. The results will be discussed in tertis of the LMO’s themselves. One should remember that this is equivalent to a discussion of the local properties of the electron density in (1). This has some consequences for the comparison of the LMO’s obtained in the CND0/2 and in the INDO approximations. For the cases considered here, we found that the essential difference lies in the treatment of the u and 71symmetries where they occur. The canonical HartreeFock orbitals are symmetry adapted with respect to possible u and n symmetry. The ER localization gives

LETTERS

15 July 1974

in this case LMO’s which are not of pure u or 71symmetry. This can be understood in terms of the relations (1) and (2). The INDO method also results in this situation. The CND0/2 scheme, however, yields LMO’s which retain the u and 71symmetries. This seems to be due to the neglect of the one-center exchange integrals. In the cases considered here the differences between the CND0/2 and INDO localized molecular orbitals are conceptual rather than real. It reflects the fact that it is possible to reproduce an axially symmetric electron density by either a sum of u and 7~ orbitals or by a sum over bent bonds. The difficulties mentioned in ref. [8] about indeterminancy for localization in CND0/2 do not appear in the systems considered here. In the following we discuss the hydrogen bond mainly in terms of INDO LMO’s. This method suffers, however, from the drawback of not having a welldefined parameter set associated with the second row atoms. Calculations were consequently carried out using both the CND0/2 and INDO schemes. For systems containing second row atoms, a modified version [ 131 of Santry’s parameters [ 141 was used.

3. Results and discussion We report here results for the hydrogen bonded dimers of HF, H20, BH,, PH,, H,S, LiH and for the hypothetical acetylene-C2 hydrogen bonded complex. The rklevant LMO’s are found in tables 1, 2 and 3. As we are interested in how well these semi-empirical LMO’s correspond to the classical “lone-pair hydrogen” picture of a hydrogen bond we give here the electronegativities [15 ] of the proton donors and acceptors involved (HF), (4.0); (H20)2 (3.5); (H2S)2 (2.5); C2H2--C, (2.5); (PH3)2 (2.1); (BH& (2.0) and (LiH), (1 .O). (a) HF dimer. As the HF dimer is well knownand has been studied throughly with both “ab initio” [ 161 and semiempirical [ 1 ] techniques, it was chosen as the system of greatest concentration. Maintaining the monomer minimum energy (INDO) F-H bond-length of 1 .OOA [ 16,171 a minimum energy for the linear dimer (I) was found for an FF distance of 2.40 A and HFH angle of 140’. This agrees well with previously 215

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Normalized included

2

CHEMICAL

LMO coefficients

for (HF)a

in the minimum

Fls

Fix

FlY

Flz

-0.81

-0.23 -0.40

PHYSICS

LETTERS

Table 1 energy geometry

Hl

15 July 1974

a). Coefficients

F2s

F2,

with absolute

F2_,,

LMOl

INDO CNDO

0.53 0.92

-0.08

LM02

INDO CNDO

0.53

-0.66 1 .oo

0.48

-0.23

LM03

INDO CNDO

0.53

0.75

0.33 1 .oo

-0.23

LM04

INDO CNDO

0.34 0.32

LM0.5

INDO CNDO

0.33 0.31

LM06

INDO CNDO

0.53 0.55

0.64

-0.53 -0.76

LM07

INDO CNDO

0.53

-0.75 1.00

-0.38

LM08

INDO CNDO

0.52 0.73

0.12

0.75 0.74

1

H

1

__-kf* 2

(1)

published results [ 16,171. The localized orbital coefficients for this structure are given in table 1. For purposes of qualitatively characterising the LMO’s, the subjective criterion that coefficients less than 0.05 are not significant was used here and in the rest of this work. From table 1 it can be seen that the classical idea of a hydrogen bond is born out in a qualitative sense, with all the lone-pairs and normal bonds appearing. It is seen that there is a significant contribution to the LMO describing the hydrogen bond from both the F2 lone-pair and the hydrogen involved in the bond. This is consistent with the classical idea that the hydrogen bond is formed by interaction of Hl with one of the F2 lone-pairs. For comparison purposes the coefficients of the LMO’s in the monomer 216

H2

lone-pair

lone-pair 0.57 0.60

0.14 0.12

_

F22

lone-pair

a) The complex is in the yz plane with Fl HlF2 along the positive for INDO and (1 .O A, 2.45 A and 135”) for CNDO/Z.

F

value less than 0.05 not

F-H

z axis. R(FH)

0.47 0.47

0.48 0.26

= 1 .O A, R(FF)

0.57 0.56 0.15 0.33

bond

0.59

D.61 F-H bond lone-pair

lone-pair -0.68 -0.62

hydrogen

= 2.40 A, angle (HFH)

bond

= 140”

are presented in table 2. For the linear structure of the dimer one finds three equivalent LMO’s each one mainly built from a lone-pair on F2 and the hydrogen atom Hl. This of course reflects the cylindrical symmetry of this structure. For a bent dimer the axial symmetry is broken which gives the situation depicted in the INDO results of table 1. The energy-localized LMO’s used here give a convenient graphic description of the formation of the hydrogen bond (fig. 1) and its properties under changes in the geometry of the molecules (fig. 2). The variations in the s-p hybridization seen in the figures are an artifact of the localization procedure used. In varying the position of the bridging hydrogen HI we found that the LMO describing the F2-H2 bond got a small contribution from Hl for smaller HI-F2 distances. In the minimum energy (INDO) geometry we found lC,, ( = 0.04 in the LMO describing the F2-H2 bond. In the linear case the coefficient increased to 0.07. This contribution increases somewhat with diminishing HI -F2 distance, For the cyclic (HF)2 dimer in the minimum energy

15 July 1974

CHEMICAL PHYSICS LETTERS

Volume 27, number 2

Table 2 Normalized LMO coefficients in the HF monomer (orientation as F2-H2

FIX

FS LMOl

INDO CNDO

0.31 0.29

LM02

INDO CNDO

0.53

LM03

INDO CNDO

0.53

LM04

INDO CNDO

0.53 0.93

FY 0.47 0.53

-0.05 0.73 1 .oo -0.68

geometry of Kollman and Allen [l] qualitatively similar results are obtained. The contribution of more than two atoms to an LMO brings up the question of the role of threecenter bonding and its connection with hydrogen in the present localized description. The presence of a three-center LMO involving the bridging hydrogen reflects the deficiences in the classical lone-pair picture of a hydrogen bond. This is well illustrated by the B2H6 molecule where the hydrogen bridges are entirely described by three-center LMO’s. We also studied the FHF- ion. Besides the usual lone-pairs on the fluorine atoms there are two additional LMO’s describing the two FH bonds, respectively. The dif-

Fz 0.56 0.53

0.73 0.71

-0.42 -0.71

0.14

-0.41

0.20 -0.26

-0.46 0.26

in the dimer table 1) H 0.60 0.61

F-H bond lone-pair lone-pair lone-pair

ferent localized pictures of B2H6 and FHF- are understandable if one remembers that the 4 valence electrons involved in the bridging orbitals pairwise are distributed according to (2) [ 181. In the (FH), dimer the situation is complicated by the presence of the second hydrogen atom and the simple arguments in the case of FHF- do not hold. The LMO picture of the (FH), dimer is thus characterized by the interaction between the hydrogen atom and the lone-pairs in the fluorine atom. In the linear structure all lone-pairs participate on an equivalent basis while in the bent form one of them dominates. The CNDO model puts all the interaction through one of the lone-pairs even in the linear geometry

0.6

20

30

RF

Fig. 1. Coefficients (INDO) for LM08 in (HF)? as function ofR(FF)for 0 = 135”.

Fig. 2. Coefficients (INDO) for LM08 in (HF)2 as function of 0 with R(FF) = 2.45 A.

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CHEMICAL PHYSICS LETTERS

15 July 1974

Table 3 Normalized coefficients for LMO’s describing hydrogen bonds. Geometries used as in text. Coefficients with absolute values less than 0.05 not included H20-HOH INDO CNDO H2S-HSH CNDO

OS

ox

0.57 0.82

* 0.71

SS LMOl LM02

0.87

H3P-HPH2

PS

CNDO

0.83

LMOl LM02

HCCH-C2 INDO CNDO BzH6 INDO CNDO LiH-LiH INDO CNDO

Sz 0.49 (0.05) Pz 0.52 (0.11)

0,

H

0.40 0.55

0.09 0.12

sd

H

S2s

S2z

0.07 0.70

0.25

-0.65

(0.07) 0.12

H

P2s

P2z

p2d

0.10 0.70

0.24

(0.12) -0.61

0.14

c3,

C4x

-0.36 -0.36

0.11 0.11

(0.07) pd -0.10 (0.17)

Cl,

C2s

C2x

H

c3,

-0.06 -0.05

0.42 0.44

0.49 0.48

0.63 0.63

0.18 0.18

Bs

I%

Bz

H

B2s

0.31 0.31

0.66 0.65

0.26 0.26

0.26 0.26

-0.35 -0.35

Li,

Li,

H

Li2,

0.40 0.41

0.36 0.37

0.68 0.68

0.35 0.36

There is only a small contribution from three-center bonding of the kind H-F-H, but none from F-H-F, (b) Hz0 dimer. Calculations were carried out for a variety of O-O distances using the minimum energy geometry of the linear system [ 1 ] . The localized description (INDO) of the hydrogen bond is dominated by two equivalent LMO’s with contributions from the bridging hydrogen and the two oxygen lonepairs, respectively. That is, we get a picture similar to the one in the HF dimer. As for the latter dimer there is also a spurious (CH = 0.03) contribution from the bridging hydrogen in the LMO for the outer HO bonds on the proton acceptor. There is no significant O-H-O contribution in the localized orbitals. Comparing the CNDO and INDO results in table 3 one again gets an illustration of the different treatments of u and 71electrons. The CNDO model 218

B2Y -0.35 -0.35

=d

B2z -0.31 -0.31

Hb -0.06 -0.05

Li2, -0.35 -0.33

gives the hydrogen bond as in table 3 and further a 2P~ lone-pair on the oxygen atom besides the usual localized lone-pairs and OH bonds. (c) H2S dimer. For reasons mentioned previously only the CNDO/2 method was used. We believe that an INDO calculation would give results analogous to the water case. The H2S dimer contains a weak hydrogen bond [19] . In table 3 we give the LMO for this bond calculated in the minimum energy geometry [ 191. The bridging hydrogen participates to a lesser extent in the outer HS bonds on the proton acceptor atom in comparison with the water system. (d) PH3 dimer. Calculations were carried out (CNDO/2) using the PH, monomer experimental geometry [20] and allowing a PH3 to approach the second along a PH bond. For a PP distance of 3.2 A

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PHYSICS

the LMO describing the hydrogen bond is given in table 3. As the PH bond is not completely localized we also show this LMO. The contribution of the bridge hydrogen in the outer PH bonds is negligibly small. (e) HCCH-Cz system. Calculations were carried out for the linear system using the CNDO minimum energy fragment geometries [ 111 (II) and C2-C3 distance

Hl-Cl=C2-H2----C3=C4 (II)

of 2.2 8. The LMO’s involving H2 are shown in table 3. In this case we of course do not expect the classical lone-pair picture. One sees that a three-center LMO is emerging like the one in the B2H6 case. One finds minor contributions of H2 (CH2 x 0.04) in the four LMO’s describing the C3-C4 molecule in the INDO model, while LMO’s from CNDO retain the u and n symmetry. (f) BH3 dimer. Calculations

on B,H, (III) were carried

LETTERS

15 July 1974

distance of 1.57 A and an Li-Li distance of 3.17 A. This is close to the energy minimum in the linear form. Except for the last three systems (e)-(g) we get the well known charge-transferance from proton acceptor molecule to proton donator molecule. For the complex (e) and (g) we find the opposite situations. In conclusion, then, we find that the localized orbitals extracted from a CNDO/2 or INDO calculation give a pictorial and consistent description of the hydrogen bonded systems discussed here. The deficiency of the electrostatic model for many hydrogen bonded systems has been apparent for several years [S] . In the present localized model the classical electfostatic lone-pair model of hydrogen bonding is complemented by non-trivial, but small contributions from adjacent A-H bond orbitals. These contributions are found for a variety of host atoms in a variety of geometrical configurations. The “classical” hydrogen bonds associated with the highly electronegative atoms, F, 0, and N are in the present model predominantly described as “lone-pair hydrogen” bonds, while the less electronegative atoms form hydrogen bonds where a three-center A-H-B bond dominates. This change in bond character can also partly be understood if one considers the number of electrons available to participate in the hydrogen bond and, in comparing two periods, the internuclear distances.

Acknowledgement out for a number of B-B distances and angles 8. In all cases the distance to the terminal hydrogens was kept at 1.19 A while that to the bridge hydrogens was kept at 1.33 8. The LMO’s involving the bridging hydrogens are presented in table 3 for R(BB) = 1.78 8. One sees there the typical three-center BHB orbitals of this compound In addition to these there are four localized BH bonds to the terminal hydrogens, but no HB-H character. (g) LiHdimer. The linear dimer was considered and for a variety of Li-Li distances a localized molecular orbital describing a nearly symmetric three-center Li-H-Li bond was found. A well-localized Li-H bond also appears. The values in table 3 are for an LiH

Acknowledgement is made to the Donors of the Petroleum Research Fund, administered by the American Chemical Society for partial support of one of us (JRS) and to the Swedish Natural Sciences Research Council for support (PL). This work was also supported by a grant of computer time from the University of Florida Computing Center.

References [l]

P.A. Kollmanand L.C. Allen, J. Am. Chem. Sot. 92 (1970) 753. [2] P.A. Kollman, J. Am. Chem. Sot. 94 (1972) 1837.

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[3 J M.S. Gordon [4] [5] [6] (71 [S] [9] [lo]

and D.E. Tallman,

CHEMICAL

Chem. Phys. Letters

PHYSICS

17

(1972) 385. R.E. Christofferson, L.L. Shipman and G.M. Maggiora, Intern. J. Quantum Chem:SS (1971) 143. S. Bratoz, in: Advances in quantum chemistry, ed. P.-O. Liiwdin (Academic Press, New York, 1967) p. 209. C. Edmiston and K. Ruedenberg, Rev. Mod. Phys. 35 (1963) 457. C. Edmiston and K. Ruedenberg, J. Chem. Phys. 43 (1965) 597. E. Switkes, R.M. Stevens, W.N. Lipscomb and M.D. Newton, J. Chem. Phys. 5 1 (1969) 2085. C. Edmiston and P. Lindner, Intern. J. Quantum Chem. 7 (1972) 309. C. Trindle and 0. Sinanoglu, J. Chem. Phys. 49 (1968) 65; W. England and M.S. Gordon, J. Am. Chem. Sot. 91 (1969) 6864.

LETTERS

[ll ] J.A. Pople and D.L. Beveridge, Approximate molecular orbital theory (McGraw-Hill, New York, 1970). [12] P.-O. tiwdin, Phys. Rev. 97 (1955) 1474. [13] J.R. Sabin, D.P. Santry and K. Weiss, J. Am. Chem. Sot. 94 (1972) 6651. [14] D.P. Santry, J. Am. Chem. Sot. 90 (1968) 3309. [ 151 L. Pauling, The nature of the chemical bond (Cornell Univ. Press, Ithaca, 1960). [16] P.A. Kollman and L.C. Allen, J. Chem. Phys. 52 (1970) 5085; W. von Niessen, Theoret. Chim. Acta 31 (1973) 297. [17] J.R. Sabin, J. Chem. Phys. 56 (1972) 45. [ 181 G.C. Pimentel and A.L. McClellan, The hydrogen bond (Freeman, San Francisco, 1960); G.C. Pimentel, J. Chem. Phys. 19 (1951) 446. [19] J.R. Sabin, J. Am. Chem. Sot. 93 (1971) 3613. [20] L.E. Sutton, ed., Tables of interatomic distances, Special Publication No. 11 (The Chemical Society, London, 1958).

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15 July 1974