Ab initio studies on hydrogen-bonded chains. III. The linear, infinite chain of hydrogen cyanide molecules

Chemical Physics 79 (1983) 211-21s North-Holland Publishing Company

211

AB INITIO STUDIES ON HYDROGEN-BONDED CHAINS. Ill. THE LINEAR, INFINITE CHAIN OF HYDROGEN CYANIDE

MOLECULES

Alfred KARPFEN

Ab initio crystal-orbital calculations have been performed on the linear. infinite chain of HCK molecules apply&g from minimal to double-zeta-plus-polarization quality. From computed potential surfaces many properties of the ground state of (HCN), could be deduced. Equilibrium structure. hydrogen-bond snrrgy. dip& moment per dipole-moment derivatives. harmonic force constants. vibrational freauencies and electronic band snucture are R&Its on the polymer are compared with monomer. dimer and crystal dam.

1. Introduction In the solid state hydrogen cyanide forms hydrogen-bonded, pyroelectric molecular crystals. In both the low-temperature (orthorhombic) and high-temperature (tetragonal) phase parallely oriented, one-dimensional hydrogen-bonded chains are the dominant structural features [ 11. Interchain distances are large. Closest distances between atoms situated on neighboring chains are well outside the range to be expected from a consideration of van der Waals radii. Because of this highly anisotropic crystal structure many properties of the HCN crystal may therefore be discussed if the one-dimensional chains are considered exclusively. Apart from the structural data many vibrational spectroscopic studies, infrared as well as Raman have been reported [2-81. Additionally. inelastic, incoherent neutron scattering has been applied in order to elucidate the phase-transition mechanism in solid HCN [9]. Association of HCN in the form of linear polymers in the liquid [ 10.1 l] and in the vapor phase [ 12,131 has already been noticed in early work. The dimer of HCN has been investigated by infrared [14.15] and microwave rotational spectroscopy [ 16- 181 in the vapor phase and by infrared spectroscopy in inert matrices [ 19-211 establishing a fully linear structure. 0301-0104/83/0000-0000/$03.00

0 North-Holland

basis sets electronic molecule. reported.

Compared to the enormous number of theoretical studies on the phenomenon of hydrogen bonding, HCN aggregates have only rarely been investigated- Ab initio studies on (HCN), and (HCN), performed at a minimal-basis-set level (STO-3G) and applying a frozen-geometry approximation for the HCN molecules equally predicted the linear structure to be the most stable configuration [22]. Calculations of the force field of (HCN), using a 4-3lG basis [23] and an optimization of the intermolecular distance in (HCN), applying a 6-31G ** basis [24] have appeared recently. A single ab initio study of the linear. infinite chain of HCN molecules has been reported up to now [25]. again at the STO-3G level and considering a few selected geometries only. Relevant for our purpose are also some investigations of the dipole moment and dipole-moment derivatives of the isolated HCN molecule [26-301 and an attempt to evaluate the effective dipole moment of an HCN molecule in external. axial fields. [31]. The latter result has amply been used in model calculations on the static-lattice theory 132.331 and dielectric 1341 properties of crystalline HCN. An interesting study of the lattice energy and lattice dynamics of HCN [35.36] based on a perturbation theory of intermolecular interactions has also been performed_

Continuing our attempts to study the influence of periodic crystalline environments on hydrogenbonded systems [37,38] we present detailed ab initio calculations on the linear, infinite chain of HCN molecules. Gaussian basis sets ranging from minimal to double~zeta-plus-polarization quality have been applied and the Hartree-Fock model was used. Several ground-state properties are reported. Within the restriction of a completely linear structure, geometry optimization of both intraand inter-molecular distances has been carried out assuming that the unit cell of the infinite polymer consists of a single HCN molecule only. From the numerically computed potential surfaces, harmonic force constants and frequencies of optically active in line vibrations are evaluated. We also discuss the dipole moment per HCN molecule and dipole-moment derivatives wii:h respect to internal coordinates. For completeness we report the electronic band structure too. Whenever possible ample comparison to monomer, dimer and crystal data is made. Differences in the general behavior between hydrogen cyanide and hydrogen fluoride polymers are pointed out. 2. Method of calculation As in the previous papers of this series [37,38] the ab initio crystal-orbital method [39-411 has been applied. In recent reviews (42-451 the current status of ab initio studies on polymeric systems is documented extensively. Trogress in the design of efficient computer programs currently allows calculations on simple polymers matching the numerical accuracy of analagous computations on molecules and clusters. We mention therefore only that in all numerical calculations reported in this work energy values are converged to at least five significant digits with respect to both lattice summations and to the number of k-points used in the various numerical integrations. In order to achieve this, one- and two-electron integrals had to be computed exactly up to second neighbors applying scheme I of ref. [46]. Remaining long-range interactions could be treated with the aid of electrostatic approximations [47] without loss of accuracy. The quality of our calculations therefore depends solely on the basis set applied.

----H

Fig.

-

c

G

N_____H

1. Structure of

-

c

N-_-_-H

3

the infinite,

_

c z

N--_-

linear chain of hydrogen

cyanide molecules.

Three different basis sets have been used which are denoted in the following in the order of increasing size as I: STO-3G [48], II: (8s4p/4s) + [5.3/3]. [49.50], and III: (lOs6pld/6slp) + [6.4,1/4-l], [49,50] with d-exponents of 0.95 and 1.0 on nitrogen and carbon, respectively. and a p-exponent of 0.75 on hydrogen. Assuming a perfectly linear configuration for the HCN chain (see fig. l), three-dimensional potential surfaces have been scanned with as internal coordinates_ and rN--,, Tc-H. rC,N, About thirty points have been computed on the

Table 1 Energy surface and dipole-momem obtained with basis set !I1

surface for (HCN),

‘c- H (bohr)

rc E N (bohr)

TN--H

E

P

(bohr)

(hartree)

(debye)

2.02

2.12

4.30

- 92.9 16438

3.7252

2.02 2.02

2.12 2.12

4.25 4.10

-92.916496 - 92.9 16564

3.7292 3.7198 3.7158

2.02

2.12

4.05

-92.916541

2.02 2.12

2.12 2.12

3.90

- 92.9 16242

3.6934

4.14

- 92.914639

3.8476

2.07

2.12 2.12

4.10 4.10

-92.916070 - 92.9 16062

3.7795 3.6606

2.02

2.12 2.22

4.14 4.14

- 92.914345 -92.910131

3.6076 3.8230

2.02

2.17

4.10

-92.915045

3.7699

2.02

2.07

4.10

- 92.914098

3.6733

2.02 2.07 2.07

4.14 4.14 4.14 4.14

- 92.906942 - 92.914550 -92.913555

1.97 1.97

2.02 2.17 2.07 2.17 2.07

- 92.914496

3.8364 3.7368 3.7120

2.07

2.12

4.14 4.24

-92.913611 -92.915945

3.7861

1.97 1.92

3.6345

3.6203

2.07

2.12

4.04

- 92.916074

3.7767

1.97

2.12

4.24

- 92.9 16034

1.97

2.12

4.04

- 92.915955

3.6716 3.6584

2.02 2.02

2.17 2.17

4.24 4.04

- 92.914967 -92.915022

3.7744 3.7656

2.02

2.07

4.24

2.02

2.07

4.04

- 92.914052 - 92.91405 1

3.6711

2.019 aJ

2.125

4.121

-92.916580

3.7282

I’

Equilibrium geometry.

3.6810

as

A. Karpjen / Ah inirio studies on h_rdro_qen- bonded chaim. III

energy surface with each of the basis sets I to III. Fits with various polynomials of second and third degree have subsequently been carried out in order to extract equilibrium geometry_ force constants Table 2 Ground-state

properties

of HCN. (HCN),,

and (HCN),

E (hartree)

HCN

(bohh

‘C-H

rc N (bohr) P (debye) m

(HCNh

Ill. - 92.90677 2.002 2.124 3.25

- .23.5 2.004 2.157 4.22 1.994 2.152 7.29

- 18.4 2.01 h’ 2.12-I I.’ 1.40 2.002 h’ 2.124 h’ 7.223

- 22.26 2.053 2.176

- 32.76 2.017 2.154

- 25.73 2.019 2.315

3.707 7.936 2.90

3.904 8.075 3.5 1

4.121 8.265 3.73

5.58 [22]

(bohr)

Frozen monomer geometries. Constrained values. Th.__c_ Primed coordinates refer to zhe Proron

derivatives

of HCN

with basis sets 1 to Ill

- 92.83342 1.992 2.153 3.20

15.46 =’ 122)

AE(kJ/mole) Q_ H (bohr) rc e N (bohr)

Table 3 Dipole-moment

as obtained

II

G _ x (bohr) p (debse)

‘) h’ c) d’

dipole-moment derivatives_ Vibrational of (HCN), have been computed using standard harmonic vibrational analysis 15 I]. frequencies

- 9 1.67526 2.022 2.179 2.45

Q_), (bohr) rc _ N (bohr) rN_- H (bohr) T&_” (bohr) ‘)

rN-_H (bohr) lattice period p (debye)

and

1

AE &J/mole)

WCN),

acceptor

and (HCN),

, Pcl=x

Table 4 Harmonic

as obtained

with basis set Ill. All values

exp.

talc.

ref. (261

ref. [28] CI

ref. 1281 SCF

this work SCF

1.06 - 0.32

1.02 - 0.34

1.34 0.55

1.11 0.57

stretching

force constants

of the linear.

infinite

chain

I 27.24 (27.36) - 0.28 ( - 0.35) 7.48 (8.17) - 0.07 0.37 0.19

~~~~; ~c=s.N--H fC-H.&H f.u-4%-H

a’ Values

in parentheses

refer to the isolated

HCN

Exp. 2.014 [52] 2.179

(Sl]

2.99 1531 - IS.4 1171 _ 6.21 “[I71

6.0-7.0

] 171

_ s.20 [I]

molecule.

HCN

6-H

213

molecule.

of HCN

molecules.

in D/.i

(HCX), cslc. this work SCF

HCN crystal cxp. ref. [6]

-.’ ‘6 1.P

12. &l 2 1.20

hll values

in mdTns/.i

II

111

23.0s (23.00) - 0.27 ( - 0.29) 6.57 (7.10) 0.022 0.23 0.19

24.63 (24.5 1) -0.17 (-0.16) 6.4s (6.96) 0.040 0.16 0.17

214

A.

Karpfea / Ab irririo studies oo hydrogen-bonded

3. Results

Table 5 Vibrational

As a representative example for the computed energy surfaces we show total energies and dipole moments per unit cell as a function of internal coordinates as obtained with the most extended basis set III in table 1. Final ground-state properties of (HCN), evaluated with basis sets I to III are collected in table 2. Hydrogen-bond energies. equilibrium geometries, and dipole moments are compared with corresponding monomer. dimer and crystal properties_ Dipole-moment derivatives with

(HCN),

respect to internal table 3. Harmonic

frequencies

of

stretching

modes

in

HCN

and

as obtained with basis set III. All values in cm-’ HCN

(HCN),

HCN crystal

talc.

exp. 1261

CdC.

esp.

3645.5 2427.5

3312 2089

3507.6 2426.7

3130 2099

and band structures)

coordinates are displayed in force constants of the HCN

chains. III

widths at the respective are presented in table 6.

[6]

equiIibrium

4. Discussion

chain are confronted with those of the isolated molecule in table 4. Vibrational frequencies for optically active modes of the polymer and monomer frequencies are shown in tabIe 5. Finally the most important band-structure data (band edges

4. I. H_vdrogen-bond

energies

Experimental

dimerization

ergies of hydrogen

cyanide

enthalpies

and

have been reported

enin

Tahlr 6 Band structure data and Mulliken gross populations of the linear. infinite chain of HCN

molecules. Energy values in eV

l

II

111

c,(O) =’

-4lS.24618

- 424.84284

- 424.25475

r,(c)

-418.24600

- 424.842265

- 424.25470

Jl ‘2(O)

0.00018 -301.31424

0.00019 - 307.6727 1

o.oOOO5 -307.11399

t2(=j

- 301.31470

- 307.67228

-307.12400(-307.00)

0.00046 -32.1695 - 32.073 1

0.00057 - 34.8 142 - 34.7671

Jz E>(O) Q(V) J, Q(O) +(-) J, %(O) es(-) As

0.0964

0.047 1

- 19.3581 - 20.5242

-21.3025 - 22.075 1

1.1661

0.7726

- 14.5335 - 12.6129

- 16.7317 - 15.4768

( - 424.22) ”

O.OOOCJl - 34.3086 - 34.2643 (-34.14) 0.0443 -21.4819 -22.1170

( - 22.06)

0.635 1 - 16.7572 - 15.7423

(-

15.81)

(-

13.79)

1.9214

1.2649

1.0149

- 12.0708 - 11.9509

- 13.9485 - 13.7796

- 14.0267 - 13.8920

A* 7

0.1119

0.1689

0.1347

9N ‘d)

7.22

7.25

7.39 (7.24)

4c

5.98

6.12

5.80 (5.92)

qti

0.79

0.63

0.81 (0.84)

Ea..(O) ,=’ %.7(=)

a’ E_(O) and E,(V) are energy values in the center and the edge of the first Brillouin zone. respectively. d, band. h’ Eigenvalues of the HCN molecule. Cl Doubly degenerate T-band. d’ Multiken populations.

is the band width of the ntb

refs. [11.13-15.171.

-23.9

kJ/mole.

Values range from - 13.8 to The most accurate one is proba-

bly due to Buxton et al. 1171 who suggested a value of A E = - 18.4 kJ/mole. Earlier theoretical results

cover about the same range. STO-3G [22]. 4-31G 1231 and 6-31G** 1241 calculations yield - 15.5, - 22.2, and - 19.7 kJ/mole. respectively. With our basis sets II and III we obtain -23.5 and - 18.4 kJ/mole in reasonable agreement with refs. [23,24]. The only theoretical trimer stabilization energy of -33.9 kJ/mole [22] is also comparable with an older experimental trimerization enthalpy of -36.4 kJ/mole. In the case of the infinitely long HCN polymer we find stabilization -32.8, and - 25.7 kJ/mole energies of -22.3. with basis sets I to III. The STO-3G value of -40.8 kJ/mole reported in ref. [25] is considerably too high. Trends in the basis-set dependence of computed hydrogen-bond energies are fully in accord with the general experience from ab initio studies on hydrogen-bonded dimers [54,55]. STO3G (basis set I) values are too small because of a substantial underestimation of dipole moments in the monomer_ Double-zeta basis sets (II) exaggerate stabilization energies mainly as a consequence of the basis-set superposition error. In the specific case of (HCN), the difference between the hydrogen-bond energies obtained with basis sets II and III is smalIer than usually observed. This is caused by a smaller dipole moment of the HCN molecule with basis set Ii. The increase of “effective hydrogen-bond energy” from the vapor-phase dimer to the infinite chain cannot be fully explained by pairwise-additive interactions with more distant molecules. Cooperative, non-additive interactions contribute a sizeable additional stabilization_ From a consideration of pan-wise-additive models for a chain of dipole molecules 1561 one can derive a 2% increase in the stabilization energy per molecule compared to the isolated dimer. With basis sets I to III we compute an increase of 44. 39, and 40% respectively_ Cooperative effects amount therefore to = 15% of the dimer stabilization in the case of (HCN),. It is interesting to note that although the HCN molecule has a much Iarger dipole moment than the hydrogen fluoride molecule (experimentally 2.99 versus 1.85 debye) non-additivities are

distinctly more important in the case of the infinite chain of hydrogen fluoride molecules 137.38.573 where the total increase in stabilization energy lies around 60-70% depending on the basis set applied.

Hydrogen bond distances in the infinite chain are systematically reduced compared to their value in the dimer. These reductions are 0.32. 0.32. and 0.28 bohr with basis sets I to 111. respectively. The most accurate experimental hydrogen bond distance (J-~___~) in (HCN), is 6.21 bohr [17]. In crystalline HCN an N--C distance of 6.02 bohr was found [l] assuming an unchanged gas-phase geometry for the HCN molecule in the crystal. The computed reduction of the intermolecular distance is therefore in reasonable agreement with the cxperimentally observed shortening of 0.19 bohr. In order to assess the importance of cooperativity in HCN chains \ve may again refer to additive models of intermolecular interactions as a guideline [56]. Pairwise-additive models would give a contraction of the intrrmolecular distance of 0.04 bohr only_ In hydrogen fluoride chains the corresponding reduction of the intermolecular distance is significantly larger. Both experimentally and theoretically [37.X57] the F---F distance is reduced by = 0.55 bohr. HCN again lies between these t\vo extremes. Relaxation of the intramolecular geometry on polymer formation is weak. The carbon-nitrogen triple bond length remains practically constant. the carbon-hydrogen distance is slightly widened by = 0.02 bohr. 4.3. Dipole momem

and dipole-momem

dericarices

An mterestmg quantity deseming further discussion is the dipole moment per unit cell. Strictly taken the dipole moment per unit cell is not uniquely defined. Its value depends on the formal choice of the unit cell [58.59]. Deciding. however. for a particular choice of the unit cell (arrangement of nuclei and basis functions) the resultant dipole moment and its derivatives with respect to nuclear coordinates are defined and may be

compared molecular models

with monomer values. In the case of crystals this choice is obvious. Many

for molecular

crystals

start out from multi-

poles and polarizabilities of isolated molecules. However. as 3 consequence of mutual polarization molecular properties” are these “gas-phase modified in the crystal. Since the electronic clouds of the constituting molecules overlap the definition of effective multipoles or polarizabilities always contains some arbitrariness because it requires necessarily a partitioning of the electronic density

in space. Within the crystal-orbital formalism multipole moments per unit cell are defined in close analogy to a Mulliken population analysis and are fixed once the position of nuclei and basis functions are defined. For recent alternative attempts to obtain dipole moments and polarizabilities in hydrogen bonded crystals see refs. [60.61]. Using a basis set comparable to our basis III, Bounds et al. [31] computed dipole moments of the HCN molecule in external. axially oriented. electric fields as a function of field strength_ Together with local-field theories they arrived at a very high value of 4.75 debye for the effective dipole moment of HCN in the orthorhombic crystal structure. thus predicting an increase of 60%. In contrast to this enormous enhancement we find from our calculations a value of 3.73 debye and hence a corresponding increase of = 15% only. Judging alone on the moderate increase of the interaction energy in the polymer, which is largely dominated by dipole-dipole interactions, the value given by Bounds et al. seems grossly overestimated. Since the interactions with neighboring molecules and hence the details of the rearrangemen! of electronic charge distributions has not been taken into account by these authors we believe that our value is more realistic. In the hydrogen fluoride chain [38] the dipole-moment enhancement was found to be 15-20% of the dipole moment of the HF monomer in the vapor phase. Dipole-moment derivatives of the HCN molecule have often been studied both experimentally [26,27] and theoretically [28-301. It is well established that, even if large basis sets are applied, a wrcng sign for ~‘c_~ is obtained within the framework of the Hartree-Fock approximation. The experimentally observed negative sign for P>_~

can only be reproduced if electron correlation is taken into account. This was demonstrated conclusively by Liu et al_ [28]. Despite this methodical deficiency of the Hartree-Fock approximation the change of dipole-moment derivatives upon polymer formation is still of interest. Experimentally, infrared intensities of both intramolecular vibrations are considerably enhanced [6] compared to gas-phase intensities. The sign of the dipole-moment derivatives in the condensed phase cannot be determined. From isotopic substitution one may. however. show that both must have the same sign. Our computed dipole-moment derivatives for (HCN), (see table 3) show a substantial enhancement over the corresponding values of the isolated molecule. Both. p> _ N and &c-n have a positive sign in the polymer. Although our computed enhancement is smaller than that observed experimentally we may conclude that this effect is reproduced qualitatively within the Hartree-Fock model and can therefore mainly be traced back to the response of HCN molecules to polarization forces. 4.4. Force constants and vibrational frequencies Trends visible in the bond-length changes of the HCN molecule upon polymerization clearly show up in computed harmonic force constants and vibrational frequencies too (see tables 4 and 5). Within a given basis set fCeNmCENand fC=N.C_H remain practically unchanged. Only fc_ n. ,-_ n is significantly lower in the polymer in line with the increase in rc_u_ The intermolecular force constantfN__H, N--H is smaller than the 0.3 mdyne/A reported by Pezolet and Savoie [7]. probably because of the too large intermolecular distance computed with basis set III. As a consequence of the overestimation of diagonal intramolecular stretching force constants within the Hartree-Fock model, computed vibrational frequencies are consistently too high. Changes from monomer to polymer are, however, quite well reproduced_ Virtually no change in the position of the C = N mode is found, whereas the C-H mode is shifted to lower frequencies. The computed shift amounts to 138 cm-‘. Experimentally the C-H frequency is reduced by 182 cm-‘. In our calculations the unit cell consisted of a single HCN molecule only.

Further coupling constants to more distant bonds could therefore not be evaluated. 4.5. Elecrrorlic baud struct~~re Due to the simple structure of the HCN chain and the large separation of molecular eigenvalues the band-structure pattern of (HCN), is not complicated. All bands have a cos(k) shape, bands do not overlap and band widths are small (see table 6). the maximum width being around 1 eV. A qualitative description of the HCN band structure has already been given in ref. [25] and our calculations agree with the interpretation given therein.

5. Conclusions Ab initio studies on infinitely long. hydrogenbonded polymers can now be performed routinely. Potential surfaces for regular polymers and the information contained therein provide valuable insight into some aspects of cooperative phenomena of intermolecular interactions which otherwise are hardly accessible. Often cluster calculations are carried out with the aim to obtain limiting values for various ground-state properties of a polymer via extrapolation. Depending on the particular property one is interested in, convergence may be rather slow. Direct use of translational symmetry is considerably more efficient and the desired results may currently be obtained at the price of a trimer or tetramer calculation. whereas a series of much larger clusters kvould have to be treated in order to guarantee a reliable extrapolation. A rather consistent picture emerged from our calculations. The hydrogen cyanide polymer studied in this work has turned out to be a system largely dominated by dipole-dipole forces. Deviations from additive behavior are modest only compared to the large non-additivities observed in hydrogen fluoride chains or. e.g.. ice. No large changes in the intramolecular geometry or in vibrational frequencies are observed on going from the vapor-phase monomer to the crys:al. AS was pointed out earlier [57] such large changes are connected with a radical change in the potential surface for proton motion between monomer and

dimer on one hand and the polymer on the other. A double-minimum structure for simultaneous proton transfer as for example dictated by symmetry in hydrogen fluoride chains is not occurring in HCN polymers because the isomer HNC has a much higher energy (= 60 kJ/mole less stable than HCN 1621). Because of the absence of such extraordinary cooperative effects in HCN. pairwise additive models for thr lattice dynamics [35.36] have been quite successful. On the example of dipole-moment derivatives we could. however. show that other expectation \*alues are appreciably changed upon polymerization Although electron-correlation effects could not be taken into account in our calculations it is very probable that correlation corrections do not alter our results substantially. With our largest basis set the dipole moment of HCN is still overestimated. whereas the polarizability is still underestimated. A good deal of error cancellation will therefore occur if electron correlation is included esplicitly in future. improved studies. The one-dimensional approximation to evaluate some crystal properties of hydrogen cyanide seems perfectly justified_

Aclinowledgement The author wishes to thank Professor P. Schuster for his interest and continuous support and Dr. A. Beyer for numerous discussions. Generous supply with computer time by the Interfakultties EDV-zentrum Wien is gratefully acknowledged.

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1361 [37] 1381 1391

chains. 111

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