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OH bonds in electric fields: electron densities and vibrational frequency shifts
17 February 1995
#KiAL LETTERS ELSEVIER
Chemical Physics Letters 233 (1995) 376-382
O-H bonds in electric fields: electron densities and vibrational frequency shifts Kersti Hermansson Department of Chemistry. Uppsala University9Box 531, S-751 21 Uppsala, Sweden
Received 6 July 1994; in final form 13 December 1994
Abstract The changes in the electron density distribution caused by varying the O-H bond length for HDO and OH- in uniform electric fields are investigated and related to the frequency shifts for the uncoupled O-H stretching vibration. Numerical integration of difference density maps, Ap= p( roH + ArOH) - p( rOH), reproduces the electronic contribution to apii”“I/ at-,, , if the integration is carried out to a distance of = 3.5 A from the 0 atom. The frequency shift A v is proportional to -E,, x [d/Ltp”=“’ (IoH) /dr&E,, , rOH) /&-OH] and electron density maps corresponding to the electronic parts of &% VP iyduced( these terms are presented. Experimentally it has been found that the OH - ion shows a frequency upshift when bound in a moderately strong field, while water molecules show a downshift. The electron density maps show why dpr”‘-“‘( roH) /drOH is positive for HDO and negative for OH -, resulting in a downshift for bound water an an upshift for bound OH-. For positive fields, a~~;dUCedl aroH is positive for both HDO and OH - and gives a downshift contribution.
1. Introduction
The electron distribution in a molecule is a sensitive probe of the strength and nature of intermolecular interaction. Many electron density investigations - experimental as well as quantum-mechanical - have been performed for molecular crystalline systems (for a recent review see Ref. [ 1 ] ) . Many of these have discussed the electron redistribution in the bound water molecule, fewer have dealt with the bound OH- ion. The main effect of intermolecular bonding on the water or hydroxide electron density is an electron flow from the H atom(s) towards 0, electron depletion at the H atoms(s) and in the oxygen lone-pair region close to the 0 atom, and electron excess further away from the oxygen nucleus [ 21. Intramolecular vibrational frequency shifts constitute another set of probes for intermolecular interac0009-2614/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDIOOO9-2614(94)01458-2
tions. Infrared and Raman measurements have been performed for a very large number of crystalline hydrates, crystalline hydroxides and aqueous solutions [ 3-51. Here only downshifts have been observed for the water molecule compared to the free-molecule frequency, while for OH- both downshifts and upshifts are observed. This is illustrated in Fig. 1, which shows the O-H frequency ranges observed experimentally for water molecules and OH- ions bound in crystals. The free-molecule and free-ion frequencies are also indicated. OH force constant shifts and frequency shifts have also been studied extensively by quantum-mechanical ab initio methods for water and OH- in small clusters and in the crystalline state (see Refs. [ 8,6], and references therein). The uncoupled anharmonic OH frequency shifts were calculated by ab initio methods in Ref. [ 93 for OH- in a field and in Ref. [ lo] for HDO.
31-l
K. Hermansson /Chemical Physics Letters 233 (1995) 376-382
HDO 1:
HDO (Qt
OH 1:
~-
OH
(9)
redistribution caused by the surroundings: the frequency shift depends on the electron density through the derivatives with respect to rOH of the permanent and induced dipole moments, i.e. dpeldro, and a~i,i’d/CkoH. In Ref. [lo] we found that for a water molecule or for an OH - ion perturbed by a uniform electrostatic field the following relation holds to a good approximation:
(1) The goal of this Letter is to relate ( 1) to electron density maps and discuss the relative magnitudes of the
Fig. 1. Illustration of observed O-H frequency ranges for water molecules and OH _ ions in crystals from IR and Raman spectroscopy. Data from Refs. [ 3,4,6] and references therein. The free molecule/ion values (at 3707 [ 31 and 3556 cm- ’ [ 71) are also shown.
contributions to the frequency shift and discuss how the differences in the electron density features between 4200
It was shown that the field components perpendicular to the vibrating OH bond have little effect on the O-H frequency shift and that the component along the vibrating OH bond gives rise to an approximately quadratic frequency versus field behaviour as the field strength is increased from a large negative (directed from H to 0) towards a large positive value (Fig. 2). The main difference between the water molecule and the OH- ion was found to be the position of the frequency maximum. The curves in Fig. 2 can be used to explain the experimental observations in Fig. 1: for water, any positive field strength gives rise to a frequency downshift, while for OH- upshifts are observed for all small and medium-strong fields, and downshifts for very strong fields only. The field strength component along the OH bond at a water H atom in an ionic crystalline hydrate is typically found in the range +0.05 to +0.15 au. In the present Letter we will discuss electron densities and frequency shifts for vibrating OH groups. The OH frequency shift is closely related to the electron distribution of the free molecule/ion and to the electron
I
I
I
I
I
I
I
I
I
HDO ,-
4000
38OO
:
?lfl~lO & r^ 8
3400
3200
3000
I
28vf
3
I
-0.06
I
I
-0.02 Electric field (a.~.)
Fig. 2. The anharmonic OH frequency plotted versus electric field strength for the uncoupled OH stretching vibration of an HDO molecule and an OH- ion in an electric field directed along the vibrating OH bond. Here and in the following figures the positive field direction is from 0 towards H (i.e. as from a distant positive point charge on the 0 side and/or a distant negative charge on the H side). RHF(DZP) calculations.
K. Hermansson /Chemical
378
water and OH - can explain the different positions the two maxima in Fig. 1.
2. Computational
Physics Letters 233 (1995) 376-382
of
details
For both HDO and OH- the vibrational mode investigated is an uncoupled O-H vibration around a fixed center-of-mass, since this is the vibration that is studied experimentally in isotope-isolated infrared and Raman experiments. The vibrational frequencies given in Fig. 2 were calculated for this vibration using the quantummechanical variational procedure described in Ref. [ 111 for an anharmonic one-dimensional potential. The dipole moment derivatives in expression ( 1) refer to the same vibrational mode. The non-vibrating O-H bond in HDO is fixed at 0.957 A. The following step-wise procedure was used to obtain the right-hand side of ( 1) : Apfre” = p’““( rou + Arou) Calculate (i> p’“( roH) and Api”“= pind( ro, + AroH) - pind( roH) =]P
in field
(rou
+
Arc,“)
tP in fie’d( rOH) -pf”(ro,)]
-pfr=b-m
+
ArmI
al. [ 131. The calculations were performed with the HONDO-8 program [ 141. Detailed analyses of basis-set and electron correlation effects on the OH- and water OH frequencies were given in Refs. [9,10]. The zero-field O-H frequencies calculated at the MP4 (ext.) level reproduce the experimental frequencies to within a few cm-’ for both OHand HDO. All relationships discussed in Section 1, including expression (1) and the condition that the frequency maximum occurs where [ 93
ar0, = dvi(rd (
+ a~i;tdd4,
dro,
ar0,
rod
_. 1
’
were found to be equally valid at the RHF( DZP) level as at the MP4 (ext.) level [ 93 . Also the electron density maps discussed in this Letter (Fig. 3) were found to be very insensitive to basis-set improvements beyond the DZP level.
l-
as the O-H bond is stretched for the free molecule/ion and for the molecule/ion in different uniform electric fields, parallel to the vibrating bond. pfm(r, rOH) is the electron distribution of the free molecule/ion and pinfield(r, rOH) is the electron distribution of the molecule/ion in a uniform electric field, both for a distance r,, of the vibrating O-H bond. Here and in most places in the following, the spatial electronic coordinate r has been omitted from the argument of p. (ii) Compute Ap’i and Ap:yd by integrating Apfr”(r)x and Apind(r)x over space, x being the coordinate in the direction of the OH bond that is being stretched. (iii) Set dpldro,-Aplbro,. (iiii) Multiply by - E,, . Most calculations presented here have been performed with a DZP basis (double-zeta quality with polarization functions on 0 and H) from Dunning and Hay [12], i.e. (9s5p)/(3s2p) for 0 and (4s)/(2s) for H, and polarization functions with exponents 0.85 for 0 and 1.00 for H. Comparisons with results from an extended, nearly saturated basis (‘ext.‘) were also made. The extended basis was ( 1 ls8p3d) /(6s5p3d) for 0 and (6s2pld)/(3s2pld) for H (see Ref. [S]) and is a slight modification of the basis of Werner et
3. Results and discussion Figs. 3a-3f show the Ap = p( rou + AToH) - p( rou) difference density in a 5.5 X 5 A’ section through the vibrating molecule/ion, when one O-H bond is stretched in such a way that the center-of-mass is kept fixed. It is the O-H bond parallel to the longest side of each map which is vibrating. The p( roH + ArOH) and p( rOH) densities have thus been overlapped in such a way that their centers-of-mass coincide, before the difference density is calculated. Solid contours mean electron excess, dashed contours electron loss, compared to the reference density. d p,, ldr,, values were computed from A pi / ArOH and Apr,;d/ArOH. The different p,, values have been calculated by numerical integration of the function p(r)x over the volumes displayed in Fig. 3 (with the third box length equal to the vertical box length, i.e. a 5.5 X 5 X 5 A” box) and gives a pu,, value which in each case differs by only a few percent from the correct value of the dipole moment component as given analytically by the Hartree-Fock program. In Figs. 3a and 3d, the function shown is Ap=pfr~(1.013 A) -pf”(0.957 h;) for OH- and HDO, respectively. In 3b and 3c, and 3e and 3f we
K. Hermansson / Chemical Physics Letters 233 (1995) 376-382
pfrec(rJ
pfre’(ri),DZP
pind(r.J - plnd(r,), E,l = -0.05, DZP
a
pind(rJ
319
- pind(rl), E,,= 0.05, DZP
b
HDO
d
e
Fig. 3. Difference density maps for the OH- ion ((a)-(c)) and HDO ((d)-(f)). RHF(DZP) calculations. The 0 atom in OH- lies to the left of the H atom. There are two black dots for each nucleus, but this is most clear for the vibrating H atom where the two black dots are 0.056 .& apart. Solid lines indicate electron excess, dashed lines electron deficiency; the zero contour has been omitted. Contour levels at + / - 0.000001, 0.00001, 0.0001, 0.001, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, . (a) and (d): Ap=p”“(l.O13 A) -pfme(0.9S7 8). (b), (c), (e) and (g): A)]. AP= [P i”fie’q1.013 A) -pfr=( 1.013 A)] - [p’“““d(O.957 A) -$=(0.957
show how the field-induced electron redistribution, which gives rise to p jnd, ,, changes with OH distance. In Fig. 3b, for example, the map refers to the OHion in a negative field of strength 0.05 au and shows: Ap ind = p’“d( 1.013 A) - p’“d( 0.957 A) = infie’d(1.013 ii) -p’“( 1.013 A)] - [p’“fie’d(0.957 [,p A) - pfree( 0.957 A) 1. 3.1. OHThe maps in Figs. 3a-3c illustrate (a) the electronic part of the free-ion dipole moment derivative, (b) the induced dipole moment derivatives for E,, = - 0.05 au and (c) the induced dipole moment derivatives for E,, = +0.05 au. The calculated d pi /dr,, value is - 2.45 D/A (see Table 1) . A negative value means a net displacement of negative charge ‘from left to right’ in the picture, and/or a net displacement of positive charge ‘from right to left’. The nuclear contribution is + 2.26 D/A, and the electronic contribution is - 4.71 D/A. This large negative value is consistent with Fig. 3a, where
we see that, upon stretching of the OH bond around a fixed center-of-mass, the O-H bond region is depleted of electron density, but the most striking effect is the depletion of electron density on the ‘left-hand side’ of the oxygen atom and the extended area of electron density increase on the ‘right-hand side’ of the H atom. The total electronic effect is an electron flow ‘from left to right’, i.e. a large negative dpi,e,/drOH value. For E,, = - 0.05 the apv/ ar,, derivative is - 3.39 D/A (Table 1) , i.e. of the same sign as the electronic contribution to dpildr,,. For E,, = + 0.05 the api,;“/ aroH derivative is + 2.82 D/A. These sign relationships are easily recognized from the electron displacements in Figs. 3b and 3c. From a simple visual inspection of the maps in Figs. 3b and 3c it is not obvious, however, why the absolute value of apil;dlaroH is about 20% larger for E,, = -0.05than for E,, = +0.05. In an attempt to establish which regions of space are more (or less) important in determining the value of Api,yd, the contribution from each ‘infinite slice’ perpendicular to x was computed by numerically integrating Apind(r)x over y and z only.
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K. Hermansson /Chemical Physics Letters 233 (1995) 376-382
The result is shown in Fig. 4, where the solid and the dashed lines refer to E,, = - 0.05 and + 0.05 au, respectively. The line for E,, = - 0.05 has been multiplied by - 1 for ease of comparison. We see that it is the features of Ap for x values between + 4 and + 5 au that account for most of the difference between apfdlarOH for E,, = - 0.05 and + 0.05 au. This is especially evident from the ‘cumulative’ dotted and dash-dotted curves in Fig. 4, which give the area of the solid and dashed curves, respectively, up to the x value in question. Referring back to Fig. 3 we indeed notice that the contour lines in ‘the rightmost parts’ of the maps stretch further out in Fig. 3b than in 3c. Let us now connect the results of Figs. 3a-3c and Table 1 with the frequency shifts shown in Fig. 1. For positive fields, the first term in expression ( 1) makes a positive contribution, i.e. a frequency upshift, because dpi/dro, is negative, and the second term makes a negative contribution, a frequency downshift, because a,u~~d/&-ou is positive. This is shown by the dashed and the dotted lines in Fig. 5a. The net result is a frequency upshift until about E,, = 0.11 au, where the two contributions cancel. For negative fields both the freemolecule dipole moment derivative and the induced dipole moment derivative contribute to creating a frequency downshift. 3.2. Water is +0.85 D/A, of which the For water d pi ldr,, nuclear contribution is + 2.13 D/A, and the electronic contribution - 1.28 D/A (Table 1) . For OH - the electronic contribution was much more negative: - 4.71 D/A. The difference is a consequence of the presence
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1
-0.02
-6
I
I
-4
-2
O-H
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I
0
2
4
6
8
x (a.u.)
Fig. 4. The contributions to Ap’,? ( = jllAp’“(r)x dr dy dz) for OH -, from ‘infinite.slices’ perpendiculartoxcomputed from numerical integration of IlAp’““(r)x dy dz for E,, = -0.05 (solid curve) and + 0.05 au (dashed curve). The E,, = - 0.05 curve has been multiplied by - 1. The dotted (E,, = -0.05) and dash-dotted (E,, = + 0.05) curves give the ‘cumulative’ value of A CL’,?up to the x value in question. The 0 and H positions for rOH= 0.957 b; and rOH= 1.013 8, are indicated at the bottom of the figure. The arrows on the horizontal line for y = 0.0 mark the left and right borders of the electron density maps in Fig. 3.
of the second OH bond in water: the electron density increase around the non-vibrating OH bond in water is counteracting the electron migration from 0 towards H as obtained for the free OH- ion (cf. Figs. 3d and 3a). The contribution from the positive d& ldro,
Table 1 The dipole moment derivative (in D/A) for the one-dimensional OH stretching vibration around a fixed center-of-mass, for an isolated and an in-field water molecule and OH- ion. The nucl and el superscripts refer to the nuclear and electronic contributions to the dipole moment derivatives (see text). DZP( RHP) level of theory. It is the underlined values that are illustrated in the maps in Fig. 3
OH-
HDO
0.00 - 0.05 0.05
-4.71 -4.71 -4.71
- 3.39 2.82
-4.71 -8.10 - 1.89
2.26 2.26 2.26
- 2.45 -5.85 0.37
0.00 - 0.05 0.05
- 1.28 - 1.28 -1.28
-2.35 -1.69
- 1.28 -4.64 -0.41
2.13 2.13 2.13
- 0.85 -2.50 2.54
381
K. Hermansson / Chemical Physics Letters 233 (1995) 376382
HDO
I
0.02 (al
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I
0.06
I
0.10
I
I
0.14 (b)
Electric field strength (a.u.)
Fig. 5. The two terms on the right-hand
0.02
side of expression
value for water is a downshift for positive fields (see the dotted line in Fig. 5b). For the OH - ion, on the other hand, the large negative value from the electronic part of d& ldro, is only partly counteracted by the positive nuclear part, and d p\ ldrou is negative. The contribution to the OH frequency shift from d pt ldro, is thus of opposite signs for water and OH -. The derivative of the induced polarization has the same sign for water and OH- but is approximately 1.5 times larger for the latter (Table 1) . This difference is completely reproduced by integrating the electron density maps, although this may be difficult to see from a visual inspection of Fig. 3. The contribution to the frequency shift from the induced dipole moment is via the term - iE,, a~j’~(E,, , rOH)laroH and just as for OH - the effect for water is a downshift for both positive and negative fields. The net effect of the two terms in expression ( 1) is a downshift for all positive fields and for all negative fields except the very small ones (Fig. 5b). This is in total agreement with the frequency versus field behaviour in Fig. I.
4. Summary In this Letter we have studied the effect of electron density redistributions on the uncoupled O-H stretch-
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0.06
I
0.10
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I9.1‘1
Electric field strength (au.)
(1). and their sum, for (a) OH- and (b) HDO.
ing vibrational frequency for HDO and OH-. Numerintegration of difference density maps, ical + Aro,) - p( roH) , accurately reproduces Ap=p(rou the electronic contribution to ap,, /&o,, if the integration is carried out to a distance of = 3.5 A from the 0 atom. The outer regions, outside the van der Waals radius of the molecule, contribute approximately half of the electronic contribution. The frequency shift for the molecule/ion in a field is a function of d~~,nuc,ldrou, d~~,e,ldro, and a~il;,de,laroH according to expression (1). The signs and relative magnitudes of the electronic contributions, for both OH - and HDO, were illustrated by electron density maps. Figs. 3a and 3d show the maps for a vibrating free ion and molecule. The rearrangement is complex in both cases but the net effect is a positive value for HDO and a negative value for A~:/Arou OH -. These different signs lead to different signs for the frequency shift originating from the term. This is the reason behind the - J% d ~kldrou opposite frequency shifts observed for an OH - ion and a water molecule in a small or moderately strong positive electric field. The induced dipole moment derivative has the same sign (positive) for OH- and HDO for all field strengths and contributes with a frequency downshift for both positive and negative fields. When the positive field is
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K. Hermansson / Chemical Physics Letters 233 (1995) 376382
large enough this term will outweigh the upshift term from the permanent dipole moment derivative for OHand the net frequency shift is then a downshift also for OH-.
Acknowledgement This work has been supported by a grant from the Swedish Natural Science Research Council. Stimulating discussions with Professor Philippe Bopp, Dr. Jargen Tegenfeldt and Dr. Lam Ojamae are gratefully acknowledged. Fig. 4 was added after a comment from a referee.
References [ l] P. Coppens, Ann. Rev. Phys. Chem. 43 (1992) 663.
[2] K.Hermansson,ActaCryst.B41 (1985) 161. [ 31 M. Falk and 0. Knop, in: Water, a comprehensive treatise, Vol. 2, ed. F. Franks (Plenum Press, New York, 1973) ch. 2. [4] H.D. Lutz, Struct. Bonding (Berlin) 69 (1988) 99. [5] P.A. Giguere, Rev. Chim. Miner. 20 (1983) 588. [6] K. Hermansson, Chem. Phys. 159 (1992) 67. [ 71 J.C. Owrutsky, N.H. Rosenbaum, L.M. Tack and R.J. Saykally, J. Chem. Phys. 83 ( 1986) 5338. [8] K. Hermansson and L. Ojam&e, Intern. J. Quant. Chem. 42 (1992) 1251. [9] K. Hermansson, J. Chem. Phys. 95 (1991) 3578. [IO] K. Hermansson, J. Chem. Phys. 99 (1993) 861. [ 111 M.J. Wojcik, J. Lindgren and J. Tegenfeldt, Chem. Phys. Letters 99 (1983) 112. [ 121 T.H. Dunning Jr. and P.J. Hay, in: Modem theoretical chemistry, Vol. 3. Methods of electronic structure theory, ed. H.F. Schaefer III (Plenum Press, New York, 1977) p. 1. [ 131 H.-J. Werner, P. Rosmus and E.-A. Reinsch, J. Chem. Phys. 79 (1983) 905. [ 141 M. Dupuis, J. Rys and H.F. King, J. Chem Phys. 65 (1976) 111; M. Dupuis, A. Farazdel, S.P. Kama and S.A. Maluendes, in: MOTECC, modem techniques in computational chemistry, ed. E. Clementi (ESCOM, Leiden, 1990).
#KiAL LETTERS ELSEVIER
Chemical Physics Letters 233 (1995) 376-382
O-H bonds in electric fields: electron densities and vibrational frequency shifts Kersti Hermansson Department of Chemistry. Uppsala University9Box 531, S-751 21 Uppsala, Sweden
Received 6 July 1994; in final form 13 December 1994
Abstract The changes in the electron density distribution caused by varying the O-H bond length for HDO and OH- in uniform electric fields are investigated and related to the frequency shifts for the uncoupled O-H stretching vibration. Numerical integration of difference density maps, Ap= p( roH + ArOH) - p( rOH), reproduces the electronic contribution to apii”“I/ at-,, , if the integration is carried out to a distance of = 3.5 A from the 0 atom. The frequency shift A v is proportional to -E,, x [d/Ltp”=“’ (IoH) /dr&E,, , rOH) /&-OH] and electron density maps corresponding to the electronic parts of &% VP iyduced( these terms are presented. Experimentally it has been found that the OH - ion shows a frequency upshift when bound in a moderately strong field, while water molecules show a downshift. The electron density maps show why dpr”‘-“‘( roH) /drOH is positive for HDO and negative for OH -, resulting in a downshift for bound water an an upshift for bound OH-. For positive fields, a~~;dUCedl aroH is positive for both HDO and OH - and gives a downshift contribution.
1. Introduction
The electron distribution in a molecule is a sensitive probe of the strength and nature of intermolecular interaction. Many electron density investigations - experimental as well as quantum-mechanical - have been performed for molecular crystalline systems (for a recent review see Ref. [ 1 ] ) . Many of these have discussed the electron redistribution in the bound water molecule, fewer have dealt with the bound OH- ion. The main effect of intermolecular bonding on the water or hydroxide electron density is an electron flow from the H atom(s) towards 0, electron depletion at the H atoms(s) and in the oxygen lone-pair region close to the 0 atom, and electron excess further away from the oxygen nucleus [ 21. Intramolecular vibrational frequency shifts constitute another set of probes for intermolecular interac0009-2614/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDIOOO9-2614(94)01458-2
tions. Infrared and Raman measurements have been performed for a very large number of crystalline hydrates, crystalline hydroxides and aqueous solutions [ 3-51. Here only downshifts have been observed for the water molecule compared to the free-molecule frequency, while for OH- both downshifts and upshifts are observed. This is illustrated in Fig. 1, which shows the O-H frequency ranges observed experimentally for water molecules and OH- ions bound in crystals. The free-molecule and free-ion frequencies are also indicated. OH force constant shifts and frequency shifts have also been studied extensively by quantum-mechanical ab initio methods for water and OH- in small clusters and in the crystalline state (see Refs. [ 8,6], and references therein). The uncoupled anharmonic OH frequency shifts were calculated by ab initio methods in Ref. [ 93 for OH- in a field and in Ref. [ lo] for HDO.
31-l
K. Hermansson /Chemical Physics Letters 233 (1995) 376-382
HDO 1:
HDO (Qt
OH 1:
~-
OH
(9)
redistribution caused by the surroundings: the frequency shift depends on the electron density through the derivatives with respect to rOH of the permanent and induced dipole moments, i.e. dpeldro, and a~i,i’d/CkoH. In Ref. [lo] we found that for a water molecule or for an OH - ion perturbed by a uniform electrostatic field the following relation holds to a good approximation:
(1) The goal of this Letter is to relate ( 1) to electron density maps and discuss the relative magnitudes of the
Fig. 1. Illustration of observed O-H frequency ranges for water molecules and OH _ ions in crystals from IR and Raman spectroscopy. Data from Refs. [ 3,4,6] and references therein. The free molecule/ion values (at 3707 [ 31 and 3556 cm- ’ [ 71) are also shown.
contributions to the frequency shift and discuss how the differences in the electron density features between 4200
It was shown that the field components perpendicular to the vibrating OH bond have little effect on the O-H frequency shift and that the component along the vibrating OH bond gives rise to an approximately quadratic frequency versus field behaviour as the field strength is increased from a large negative (directed from H to 0) towards a large positive value (Fig. 2). The main difference between the water molecule and the OH- ion was found to be the position of the frequency maximum. The curves in Fig. 2 can be used to explain the experimental observations in Fig. 1: for water, any positive field strength gives rise to a frequency downshift, while for OH- upshifts are observed for all small and medium-strong fields, and downshifts for very strong fields only. The field strength component along the OH bond at a water H atom in an ionic crystalline hydrate is typically found in the range +0.05 to +0.15 au. In the present Letter we will discuss electron densities and frequency shifts for vibrating OH groups. The OH frequency shift is closely related to the electron distribution of the free molecule/ion and to the electron
I
I
I
I
I
I
I
I
I
HDO ,-
4000
38OO
:
?lfl~lO & r^ 8
3400
3200
3000
I
28vf
3
I
-0.06
I
I
-0.02 Electric field (a.~.)
Fig. 2. The anharmonic OH frequency plotted versus electric field strength for the uncoupled OH stretching vibration of an HDO molecule and an OH- ion in an electric field directed along the vibrating OH bond. Here and in the following figures the positive field direction is from 0 towards H (i.e. as from a distant positive point charge on the 0 side and/or a distant negative charge on the H side). RHF(DZP) calculations.
K. Hermansson /Chemical
378
water and OH - can explain the different positions the two maxima in Fig. 1.
2. Computational
Physics Letters 233 (1995) 376-382
of
details
For both HDO and OH- the vibrational mode investigated is an uncoupled O-H vibration around a fixed center-of-mass, since this is the vibration that is studied experimentally in isotope-isolated infrared and Raman experiments. The vibrational frequencies given in Fig. 2 were calculated for this vibration using the quantummechanical variational procedure described in Ref. [ 111 for an anharmonic one-dimensional potential. The dipole moment derivatives in expression ( 1) refer to the same vibrational mode. The non-vibrating O-H bond in HDO is fixed at 0.957 A. The following step-wise procedure was used to obtain the right-hand side of ( 1) : Apfre” = p’““( rou + Arou) Calculate (i> p’“( roH) and Api”“= pind( ro, + AroH) - pind( roH) =]P
in field
(rou
+
Arc,“)
tP in fie’d( rOH) -pf”(ro,)]
-pfr=b-m
+
ArmI
al. [ 131. The calculations were performed with the HONDO-8 program [ 141. Detailed analyses of basis-set and electron correlation effects on the OH- and water OH frequencies were given in Refs. [9,10]. The zero-field O-H frequencies calculated at the MP4 (ext.) level reproduce the experimental frequencies to within a few cm-’ for both OHand HDO. All relationships discussed in Section 1, including expression (1) and the condition that the frequency maximum occurs where [ 93
ar0, = dvi(rd (
+ a~i;tdd4,
dro,
ar0,
rod
_. 1
’
were found to be equally valid at the RHF( DZP) level as at the MP4 (ext.) level [ 93 . Also the electron density maps discussed in this Letter (Fig. 3) were found to be very insensitive to basis-set improvements beyond the DZP level.
l-
as the O-H bond is stretched for the free molecule/ion and for the molecule/ion in different uniform electric fields, parallel to the vibrating bond. pfm(r, rOH) is the electron distribution of the free molecule/ion and pinfield(r, rOH) is the electron distribution of the molecule/ion in a uniform electric field, both for a distance r,, of the vibrating O-H bond. Here and in most places in the following, the spatial electronic coordinate r has been omitted from the argument of p. (ii) Compute Ap’i and Ap:yd by integrating Apfr”(r)x and Apind(r)x over space, x being the coordinate in the direction of the OH bond that is being stretched. (iii) Set dpldro,-Aplbro,. (iiii) Multiply by - E,, . Most calculations presented here have been performed with a DZP basis (double-zeta quality with polarization functions on 0 and H) from Dunning and Hay [12], i.e. (9s5p)/(3s2p) for 0 and (4s)/(2s) for H, and polarization functions with exponents 0.85 for 0 and 1.00 for H. Comparisons with results from an extended, nearly saturated basis (‘ext.‘) were also made. The extended basis was ( 1 ls8p3d) /(6s5p3d) for 0 and (6s2pld)/(3s2pld) for H (see Ref. [S]) and is a slight modification of the basis of Werner et
3. Results and discussion Figs. 3a-3f show the Ap = p( rou + AToH) - p( rou) difference density in a 5.5 X 5 A’ section through the vibrating molecule/ion, when one O-H bond is stretched in such a way that the center-of-mass is kept fixed. It is the O-H bond parallel to the longest side of each map which is vibrating. The p( roH + ArOH) and p( rOH) densities have thus been overlapped in such a way that their centers-of-mass coincide, before the difference density is calculated. Solid contours mean electron excess, dashed contours electron loss, compared to the reference density. d p,, ldr,, values were computed from A pi / ArOH and Apr,;d/ArOH. The different p,, values have been calculated by numerical integration of the function p(r)x over the volumes displayed in Fig. 3 (with the third box length equal to the vertical box length, i.e. a 5.5 X 5 X 5 A” box) and gives a pu,, value which in each case differs by only a few percent from the correct value of the dipole moment component as given analytically by the Hartree-Fock program. In Figs. 3a and 3d, the function shown is Ap=pfr~(1.013 A) -pf”(0.957 h;) for OH- and HDO, respectively. In 3b and 3c, and 3e and 3f we
K. Hermansson / Chemical Physics Letters 233 (1995) 376-382
pfrec(rJ
pfre’(ri),DZP
pind(r.J - plnd(r,), E,l = -0.05, DZP
a
pind(rJ
319
- pind(rl), E,,= 0.05, DZP
b
HDO
d
e
Fig. 3. Difference density maps for the OH- ion ((a)-(c)) and HDO ((d)-(f)). RHF(DZP) calculations. The 0 atom in OH- lies to the left of the H atom. There are two black dots for each nucleus, but this is most clear for the vibrating H atom where the two black dots are 0.056 .& apart. Solid lines indicate electron excess, dashed lines electron deficiency; the zero contour has been omitted. Contour levels at + / - 0.000001, 0.00001, 0.0001, 0.001, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, . (a) and (d): Ap=p”“(l.O13 A) -pfme(0.9S7 8). (b), (c), (e) and (g): A)]. AP= [P i”fie’q1.013 A) -pfr=( 1.013 A)] - [p’“““d(O.957 A) -$=(0.957
show how the field-induced electron redistribution, which gives rise to p jnd, ,, changes with OH distance. In Fig. 3b, for example, the map refers to the OHion in a negative field of strength 0.05 au and shows: Ap ind = p’“d( 1.013 A) - p’“d( 0.957 A) = infie’d(1.013 ii) -p’“( 1.013 A)] - [p’“fie’d(0.957 [,p A) - pfree( 0.957 A) 1. 3.1. OHThe maps in Figs. 3a-3c illustrate (a) the electronic part of the free-ion dipole moment derivative, (b) the induced dipole moment derivatives for E,, = - 0.05 au and (c) the induced dipole moment derivatives for E,, = +0.05 au. The calculated d pi /dr,, value is - 2.45 D/A (see Table 1) . A negative value means a net displacement of negative charge ‘from left to right’ in the picture, and/or a net displacement of positive charge ‘from right to left’. The nuclear contribution is + 2.26 D/A, and the electronic contribution is - 4.71 D/A. This large negative value is consistent with Fig. 3a, where
we see that, upon stretching of the OH bond around a fixed center-of-mass, the O-H bond region is depleted of electron density, but the most striking effect is the depletion of electron density on the ‘left-hand side’ of the oxygen atom and the extended area of electron density increase on the ‘right-hand side’ of the H atom. The total electronic effect is an electron flow ‘from left to right’, i.e. a large negative dpi,e,/drOH value. For E,, = - 0.05 the apv/ ar,, derivative is - 3.39 D/A (Table 1) , i.e. of the same sign as the electronic contribution to dpildr,,. For E,, = + 0.05 the api,;“/ aroH derivative is + 2.82 D/A. These sign relationships are easily recognized from the electron displacements in Figs. 3b and 3c. From a simple visual inspection of the maps in Figs. 3b and 3c it is not obvious, however, why the absolute value of apil;dlaroH is about 20% larger for E,, = -0.05than for E,, = +0.05. In an attempt to establish which regions of space are more (or less) important in determining the value of Api,yd, the contribution from each ‘infinite slice’ perpendicular to x was computed by numerically integrating Apind(r)x over y and z only.
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K. Hermansson /Chemical Physics Letters 233 (1995) 376-382
The result is shown in Fig. 4, where the solid and the dashed lines refer to E,, = - 0.05 and + 0.05 au, respectively. The line for E,, = - 0.05 has been multiplied by - 1 for ease of comparison. We see that it is the features of Ap for x values between + 4 and + 5 au that account for most of the difference between apfdlarOH for E,, = - 0.05 and + 0.05 au. This is especially evident from the ‘cumulative’ dotted and dash-dotted curves in Fig. 4, which give the area of the solid and dashed curves, respectively, up to the x value in question. Referring back to Fig. 3 we indeed notice that the contour lines in ‘the rightmost parts’ of the maps stretch further out in Fig. 3b than in 3c. Let us now connect the results of Figs. 3a-3c and Table 1 with the frequency shifts shown in Fig. 1. For positive fields, the first term in expression ( 1) makes a positive contribution, i.e. a frequency upshift, because dpi/dro, is negative, and the second term makes a negative contribution, a frequency downshift, because a,u~~d/&-ou is positive. This is shown by the dashed and the dotted lines in Fig. 5a. The net result is a frequency upshift until about E,, = 0.11 au, where the two contributions cancel. For negative fields both the freemolecule dipole moment derivative and the induced dipole moment derivative contribute to creating a frequency downshift. 3.2. Water is +0.85 D/A, of which the For water d pi ldr,, nuclear contribution is + 2.13 D/A, and the electronic contribution - 1.28 D/A (Table 1) . For OH - the electronic contribution was much more negative: - 4.71 D/A. The difference is a consequence of the presence
I
1
-0.02
-6
I
I
-4
-2
O-H
I
I
I
I
0
2
4
6
8
x (a.u.)
Fig. 4. The contributions to Ap’,? ( = jllAp’“(r)x dr dy dz) for OH -, from ‘infinite.slices’ perpendiculartoxcomputed from numerical integration of IlAp’““(r)x dy dz for E,, = -0.05 (solid curve) and + 0.05 au (dashed curve). The E,, = - 0.05 curve has been multiplied by - 1. The dotted (E,, = -0.05) and dash-dotted (E,, = + 0.05) curves give the ‘cumulative’ value of A CL’,?up to the x value in question. The 0 and H positions for rOH= 0.957 b; and rOH= 1.013 8, are indicated at the bottom of the figure. The arrows on the horizontal line for y = 0.0 mark the left and right borders of the electron density maps in Fig. 3.
of the second OH bond in water: the electron density increase around the non-vibrating OH bond in water is counteracting the electron migration from 0 towards H as obtained for the free OH- ion (cf. Figs. 3d and 3a). The contribution from the positive d& ldro,
Table 1 The dipole moment derivative (in D/A) for the one-dimensional OH stretching vibration around a fixed center-of-mass, for an isolated and an in-field water molecule and OH- ion. The nucl and el superscripts refer to the nuclear and electronic contributions to the dipole moment derivatives (see text). DZP( RHP) level of theory. It is the underlined values that are illustrated in the maps in Fig. 3
OH-
HDO
0.00 - 0.05 0.05
-4.71 -4.71 -4.71
- 3.39 2.82
-4.71 -8.10 - 1.89
2.26 2.26 2.26
- 2.45 -5.85 0.37
0.00 - 0.05 0.05
- 1.28 - 1.28 -1.28
-2.35 -1.69
- 1.28 -4.64 -0.41
2.13 2.13 2.13
- 0.85 -2.50 2.54
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K. Hermansson / Chemical Physics Letters 233 (1995) 376382
HDO
I
0.02 (al
I
I
I
0.06
I
0.10
I
I
0.14 (b)
Electric field strength (a.u.)
Fig. 5. The two terms on the right-hand
0.02
side of expression
value for water is a downshift for positive fields (see the dotted line in Fig. 5b). For the OH - ion, on the other hand, the large negative value from the electronic part of d& ldro, is only partly counteracted by the positive nuclear part, and d p\ ldrou is negative. The contribution to the OH frequency shift from d pt ldro, is thus of opposite signs for water and OH -. The derivative of the induced polarization has the same sign for water and OH- but is approximately 1.5 times larger for the latter (Table 1) . This difference is completely reproduced by integrating the electron density maps, although this may be difficult to see from a visual inspection of Fig. 3. The contribution to the frequency shift from the induced dipole moment is via the term - iE,, a~j’~(E,, , rOH)laroH and just as for OH - the effect for water is a downshift for both positive and negative fields. The net effect of the two terms in expression ( 1) is a downshift for all positive fields and for all negative fields except the very small ones (Fig. 5b). This is in total agreement with the frequency versus field behaviour in Fig. I.
4. Summary In this Letter we have studied the effect of electron density redistributions on the uncoupled O-H stretch-
I
I
I
0.06
I
0.10
I
I9.1‘1
Electric field strength (au.)
(1). and their sum, for (a) OH- and (b) HDO.
ing vibrational frequency for HDO and OH-. Numerintegration of difference density maps, ical + Aro,) - p( roH) , accurately reproduces Ap=p(rou the electronic contribution to ap,, /&o,, if the integration is carried out to a distance of = 3.5 A from the 0 atom. The outer regions, outside the van der Waals radius of the molecule, contribute approximately half of the electronic contribution. The frequency shift for the molecule/ion in a field is a function of d~~,nuc,ldrou, d~~,e,ldro, and a~il;,de,laroH according to expression (1). The signs and relative magnitudes of the electronic contributions, for both OH - and HDO, were illustrated by electron density maps. Figs. 3a and 3d show the maps for a vibrating free ion and molecule. The rearrangement is complex in both cases but the net effect is a positive value for HDO and a negative value for A~:/Arou OH -. These different signs lead to different signs for the frequency shift originating from the term. This is the reason behind the - J% d ~kldrou opposite frequency shifts observed for an OH - ion and a water molecule in a small or moderately strong positive electric field. The induced dipole moment derivative has the same sign (positive) for OH- and HDO for all field strengths and contributes with a frequency downshift for both positive and negative fields. When the positive field is
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K. Hermansson / Chemical Physics Letters 233 (1995) 376382
large enough this term will outweigh the upshift term from the permanent dipole moment derivative for OHand the net frequency shift is then a downshift also for OH-.
Acknowledgement This work has been supported by a grant from the Swedish Natural Science Research Council. Stimulating discussions with Professor Philippe Bopp, Dr. Jargen Tegenfeldt and Dr. Lam Ojamae are gratefully acknowledged. Fig. 4 was added after a comment from a referee.
References [ l] P. Coppens, Ann. Rev. Phys. Chem. 43 (1992) 663.
[2] K.Hermansson,ActaCryst.B41 (1985) 161. [ 31 M. Falk and 0. Knop, in: Water, a comprehensive treatise, Vol. 2, ed. F. Franks (Plenum Press, New York, 1973) ch. 2. [4] H.D. Lutz, Struct. Bonding (Berlin) 69 (1988) 99. [5] P.A. Giguere, Rev. Chim. Miner. 20 (1983) 588. [6] K. Hermansson, Chem. Phys. 159 (1992) 67. [ 71 J.C. Owrutsky, N.H. Rosenbaum, L.M. Tack and R.J. Saykally, J. Chem. Phys. 83 ( 1986) 5338. [8] K. Hermansson and L. Ojam&e, Intern. J. Quant. Chem. 42 (1992) 1251. [9] K. Hermansson, J. Chem. Phys. 95 (1991) 3578. [IO] K. Hermansson, J. Chem. Phys. 99 (1993) 861. [ 111 M.J. Wojcik, J. Lindgren and J. Tegenfeldt, Chem. Phys. Letters 99 (1983) 112. [ 121 T.H. Dunning Jr. and P.J. Hay, in: Modem theoretical chemistry, Vol. 3. Methods of electronic structure theory, ed. H.F. Schaefer III (Plenum Press, New York, 1977) p. 1. [ 131 H.-J. Werner, P. Rosmus and E.-A. Reinsch, J. Chem. Phys. 79 (1983) 905. [ 141 M. Dupuis, J. Rys and H.F. King, J. Chem Phys. 65 (1976) 111; M. Dupuis, A. Farazdel, S.P. Kama and S.A. Maluendes, in: MOTECC, modem techniques in computational chemistry, ed. E. Clementi (ESCOM, Leiden, 1990).