Periodic Trends in Bond Energies: A Density Functional Study - ACS

Chapter 20

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Periodic Trends in Bond Energies: A Density Functional Study Tom Ziegler Department of Chemistry, University of Calgary, Calgary, Alberta T2N 1N4, Canada

Density functional theory (DFT) makes it possible to obtain reasonable (±25 kJ/mol) estimates of bond energies for compounds involving all elements in the periodic table. This account assesses the accuracy of thermochemical data obtained by D F T and rationalises some of the periodic trends obtained from systematic studies involving homologous series of compounds.

It is now possible to calculate bond energies with reasonable accuracy by the aid of quantum mechanical methods. This account reviews some of the results obtained by methods based on density functional theory (DFT) (1). The data available for A-B bond energies exhibit interesting periodic trends as A (or B) changes position in the periodic table. The origin of these periodic trends will be discussed in terms of fundamental concepts based on the Pauli exclusion principle and Einstein's special theory of relativity. The exposition will progress from single- to multiple- bonded systems with special emphasis on transition metal compounds. This-account is not exhaustive and references should be made to previous D F T reviews (2) as well as the excellent study by Frenking et al. (2c) based on ab initio methods and other DFT investigations in this volume . The Single Bond Main Group Compounds. Table I displays calculated and experimental X - C H 3 bond energies (3a) for the halogen series X= F,Cl,Br, I. We note that the DFT based local density approximation (lb) (LDA) overestimates bond energies, a fact that is well established by now (la). The gradient corrected BP86 scheme (4) affords on the other hand estimates that are within 5 kcal/mol of experiment. Both D F T based methods include to some degree electron correlation. Among the ab initio methods (5), Hartree Fock (HF) - without correlation- and the second order M0ller-Plesset perturbation method (MP2) -with some correlation - underestimate the bond energies. The agreement with experiment is not improved by going to higher order of perturbation in the MP4 scheme. For this series the gradient corrected D F T theory (BP86) affords the best fit to experiment. In most cases BP86 supplies better bond energies than the HF,MP2 and MP4 methods.

© 1998 American Chemical Society

In Computational Thermochemistry; Irikura, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1905.

369

COMPUTATIONAL THERMOCHEMISTRY

370

Table I. Calculated and Experimental X - C H 3 Bond Enthalpies A H Q (kj/mol), (X=F, CI, Br, I). 2

Halogen (X)

LDA

BP86

HF

MP2

MP4

Exp

F CI Br I

578.5 4243 357.4 297.6

469.0 338.1 278.4 222.3

250.5 195.6 142.1 89.0

437.3 327.3 270.9 212.8

420.5 316.8 261.2 205.2

465.2 342.3 286.7 232.8

Overlaps Energy Gap 0.26 7.5 0.34 3.8 3.0 0.35 0.36 2.1 c

a

Ref. 3a. Ôverlap between n p and M , see 1 . Energy gap (eV) between np and M , see 1. Ref. 3b 0


d

a

a

0

It follows from Table I that the X - C H 3 bond decreases in strength from fluorine towards iodine. This trend can be understood by observing that the X - C H 3 linkage is established by the interaction between the singly occupied n p and M orbitals, 1. In this interaction density is transferred from M of the C H 3 fragment - with the higher energy - to n p of the halogen atom with the lower energy, 1. The transfer is most favourable for fluorine with the n p orbital of lowest energy, Table I. In a more covalent bond the overlap might have been trend setting. However, is seen to run counter to what one would expect if it was the determining factor for the X - C H 3 bond strength, Table I. G

0

G

0

c

a

0

a

0

1

M ,

X

9

0

|

*

(n-l) 1 2 The increase in the energy of the halogen based np orbital towards higher n is a consequence of the intra-atomic Pauli core repulsion. (6). As more and more p-type core orbitals are added to the halogen (2p,3p, and 4p in the case of iodine), the n p valence orbital develops a nodal structure in the core region and expands in the valence region - in both cases in order to obey the Pauli exclusion principle which requires core and valence electrons of like spin not to be found at the same position. The expansion in the valence region, and the development of a nodal structure in the core region, raises the energy by respectively reducing the nuclear attraction and increasing the kinetic energy (6). The expansion of np due to the intra-atomic Pauli core repulsion is also responsible for the increase in the overlap towards heavier halogens. There is one more factor of importance for the trend in the X - C H 3 bond energies. As more and more core orbitals are added to the halogen, these occupied orbitals will interact repulsively with M , 2, and destabilize the X - C H 3 bond progressively from fluorine towards iodine. The destabilization is again a consequence of the Pauli exclusion principle. Essentially (2a), M will develop a nodal structure in the halogen core region similar to that of np in order to exclude core and valence electrons of like spin from the same region in X - C H 3 . The nodal structure will in turn increase the kinetic energy. The destabilizing interaction in 2 is termed inter-atomic Pauli core repulsion. (6). The importance of intra- and inter- atomic Pauli repulsion for periodic trends in thermochemistry has been discussed extensively by Kutzelnigg (6). Po

G

a

G

0

0

a

G

G

Table II. CI-MH3 Bond Dissociation Enthalpies* A H Q (kj/mol), (M=C,Si,Ge,Sn). Bond

BP86

HF

MP2

MP4

G2

Exp.

d

Energy

1

Overlap*

GapP

C-Cl Si-Cl Ge-Cl Sn-Cl a

338.1 441.4 402.1 391.2

195.6

327.3

b

346.9 455.6

348.6 472.3

0.34 0.34 0.33 0.33

3.8 4.6 4.7 4.9

c

Ref. 7a. O verlap between n p and M , see 1. Energy gap (eV) between nj\j and M , see 1. Ref.7b a

d

316.8 437.7

a


a

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Periodic Trends in Bond Energies

Table II deals with the C I - M H 3 bond where we now fix the halogen part at chlorine and vary the group-14 component from carbon to tin. The limited experimental data set indicates again that BP86 provides a better fit to experiment than HF,MP2 and M P 4 . On the other hand, the highly accurate G2 ab initio scheme (8) fares somwhat better than BP86. It is interesting to note that carbon forms a weaker bond to chlorine than the higher group-14 homologues. This is in line with the fact that the energy gap between n p and M is smallest for carbon. For the other elements, both the energy gap and the overlap are quite similar. The decrease in the C I - M H 3 bond strength from M=Si to M=Sn is instead set by the inter atomic Pauli core repulsion (7a). That is, the 3 p valence orbital of chlorine interacts repulsively with the M-core orbitals of the M H 3 fragment. The final main group example deals with the homopolar H 3 M - M H 3 (M=C,Si, Ge,Sn,Pb) molecules, Table III, where two M H 3 fragments are joined together. In this case both BP86 theory (9) and experiment exhibit a decrease in the bond strength from carbon to lead. The trend setting factor here is not the overlap (9) but rather the expected increase in the inter-atomic Pauli core repulsion as more and more core orbitals are added to the group-14 element. Thus, in this case M of one M H 3 fragment interacts repulsively with core orbitals on M of the opposite fragment. a

a

a

a


a

G

0

H 3 M - M H 3 Bond Dissociation Enthalpies

Table III.

3

A H Q (kJ/mol), (M=C,Si,Ge,Sn)

Bond

C-C

Si-Si

Ge-Ge

Sn-Sn

Pb-Pb

Method BP86 Exp .

363 367

293 321

257 276

220

185

a

a

Ref. 9a.

Coinage Metal Compounds. Dimers and hydrides of the coinage metals have been the subject of numerous studies (10) as they are among the simplest metal-metal and metal-ligand bonded systems. Table IV gives calculated M - H and M - M (M=Cu,Ag,Au) bond energies based on the non-relativistic BP86 scheme (BP86-NR) presented in the previous section as well as a quasi-relativistic extension (BP86-QR) due to Snijders et al. (1 lb,c). We note that the M - M and M - H bond energies decrease from copper to gold in the non-relativistic case, in disagreement with experiment. The decrease can largely be attributed to inter-atomic Pauli core repulsion between ISH (or ns\i) and the growing number of core orbitals on the opposite metal centre(10b,c). Table-IV. Electronic M - H and M 2 Bond Dissociation Enthalpies (kJ/mol) 3

Compound

CuH

AgH

AuH

CU2

Ag2

Au2

279.2 287.1 276 ±10

222.0 247.8 230 ±10

218.6 322.7 309±15

199.0 210.7 188±10

145.0 163.0 155±10

137.5 222.0 217±10

Method BP86-NR BP86-QR Exp . b

a

b

Ref. 11a. Ref. l i d

Adding relativistic effects (BP86-QR) substantially strengthens the Au-H and Au-Au bonds to the extent where they become the strongest in each of the homologous series. Instead silver is seen to form the weakest bond, in agreement with experiment, Table IV. The origin of the relativistic bond stabilisation - and bond contraction- has been discussed previously (10). Essentially, valence electrons in sorbitals of heavy elements can obtain high instantaneous velocities near the nuclei. The high velocities will increase the mass of the electron. The electron mass increase will in turn diminish the inter-atomic core Pauli repulsion by reducing the kinetic energy (10b,c). The effect is most pronounced for gold since its larger nuclear charge allows for the highest instantaneous velocities close to the nucleus. In Computational Thermochemistry; Irikura, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1905.


372

Transition Metal Compounds. The coinage metal compounds discussed above are unique in that ns is the dominant valence bonding orbital on the metal. This results in spectacular relativistic effects since valence ns electrons can come close to the nucleus and thus acquire high instantaneous (relativistic) velocities, at least in the case of gold. Most other 'real' transition metals (12) primarily make use of d-orbitals in bond formation, 3. Figure 1 displays calculated (13a ) M C I 3 - L bond energies, where M represents the group-4 triad M=Ti,Zr, and Hf of early transition metals and L runs trough the series L= PH2,SiH3,H,CH3,SH,NH2,CN,OCH3,OH of 'one'electron ligands. The C I 3 M - L linkage is established by the interaction between the singly occupied L and M orbitals, 3, in a way similar to 1. In this interaction, density is transferred from M of higher energy to the L ligand orbital of lower energy. For a given metal the C I 3 M - L bond energy increases with the energy gap between M and L . Thus the more electronegative ligands NH2,OCH3 and O H form the stronger bonds. The C I 3 M - L bonds are stabilized to some degree by a secondary jr-interaction, 4, involving occupied Ti-type ligand lonepairs and empty M metal orbitals. The latter are unoccupied for the early electron poor transition metals.


0

a

G

a

G

G

n

Figure 1. Calculated bond energies for MCI3-L and 5 L Co(CO)4-L The C I 3 M - L bond energies are seen to increase in going from the 3d-member titanium to the 5d-member hafnium, with the largest jump between titanium and zirconium. The most important factor here is the inter-atomic Pauli core repulsion involving L and the core-like ns,np metal orbitals with n=3,4,5 for Ti,Zr, and Hf respectively, 5. In the case of titanium, the 3s and 3p orbitals are of the same radial extent as the 3d orbital M . Hence a sizable bonding interaction, 3, for titanium will also result in strong inter-atomic Pauli core repulsions, 5. For zirconium and hafnium, the d-type M orbital is more diffuse than the ns and np core-like orbitals, resulting in less destabilization, 5. The d-orbital of titanium is more compact relative to ns and np than the corresponding nd orbitals of zirconium and hafnium, as it is free from intraatomic Pauli core repulsion due to d-orbitals with a lower n quantum number. Also shown in Figure 1 are the corresponding L-Co(CO)4 bond energies for the late transition metal cobalt. We note for hydrogen that the Co-H energy is somewhat smaller than that of the group-4 metals. This is so since the M orbital is of lower energy at the end of the transition series where the effective charge is largest. Thus the electron transfer in 3 is less favorable. For many of the other ligands we note that the C o - L bond energies are substantially lower than what one would expect from considering the difference in strength between the Co-H and the T i - H bonds. In these cases the secondary ^-interaction, 4, has a large destabilizing influence on the Co-L bond as the M d-orbital now is occupied and involved in a four-electron two orbital destabilizing interaction with L^. The only ligand somewhat out-of-line is C N since it has an additional L^* orbital capable of accepting density from M ^ . G

a

a

a

a

n


20.

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373


300.

kJ mol 250.


200.

150.

100.

50.

Th

U

Tl

CI M-R

Zr

CI3M-R

3

Hf

Mn

Ir

Tc

R-M(CO)

R-M(CO)

5

H

C H

3

i

4

Figure 2. A comparison of M - H and M-CH3 bond energies for different metals It stands to reason that metal-hydrogen and metal-alkyl bond energies are among the most important thermochemical parameters in organometallic chemistry. Figure 2 displays trends in calculated (13b) M - H and M - C H 3 energies for early as well as late transition metals. We note first of all that both M - H and M - C H 3 bond energies increase down a triad of metals on account of the decrease in the inter-atomic Pauli core repulsions, 5, already discussed in connection with Figure 1. Also M - H bonds are seen to be stronger than M - C H 3 bonds for electron rich late to middle transition metals as the acH3 bonding orbitals interact with occupied d orbitals on the metal to destabilize the M - C H 3 bond compared to the M - H linkage where such interactions are impossible. On the other hand,, for early transition metals M - H and M - C H 3 bonds are of similar strength since the d^ orbitals now are empty and the interaction between GCH3 and d stabilising (13b). In fact, the stabilizing interaction makes in some cases the M - C H 3 bond stronger, Figure 2. n

n

M+

I —• H 6a

CH M+

I

-CH

3

H 6b

3

CH

3

CH

3

^CH

3

Xhb 6c

The fact that M - H bonds are stronger than M-alkyl bonds for late to middle transition metals, Figure 2, has profound implications for elementary reaction steps in organometallic chemistry (13b, 14). Thus, H2 adds readily to a metal centre, 6a, as the two M - H bonds gained are stronger than a single H-H bond (400 kJ/mol). On the other hand a C - C alkyl bond never adds to a metal centre, 6c, as the C-C bond broken (400 kJ/mol) is stronger than ihe two M - C bonds formed. The addition of a C - H bond , 6b, is an intermediate case observed in a few cases. The migration of an alkyl group from a metal centre to the carbon of a ciscarbonyl is quite a facile process, 7a, as the M-alkyl bond is relatively weak. On the other hand, the corresponding migration of a hydrogen, 7b, is hardly ever observed due to the strength of the M - H bond.



374

CH

3

/

co

,CH

H 3

ji-

7a


/

co-X-

M— c

O

7b

The calculated data displayed i n Figures 1 and 2 were based (13) on an earlier and less accurate gradient corrected D F T scheme than B P 8 6 . However, the trends depicted are not likely to change with the introduction of more recent D F T methods. Direct validation of recent D F T methods as w e l l as ab initio schemes (12a-d) have been carried out on "bare metal" complexes i n which just one ligand is attached to a metal centre i n a single a - b o n d . Dissociation energies for these coordinatively unsaturated systems have been measured with high accuracy (12f-g) i n the gas phase for many systems. F o r 'normal' coordinatively saturated M L transition metal complexes average bond dissociation energies, M L - » M + n L - n A H , are available for a number of systems. However, accurate experimental estimates of the chemically more interesting first ligand dissociation energy, M L -> M L _ i + L - A H i , is only available for a few a - b o n d s (12f). W e compare i n Table V calculated (14) and experimental first M - H and M - a l k y l bond dissociation energies. W e note that the agreement with experiment is within 25 kJ/mol, which is the mean error we currently attribute to the calculation of M - L single bonds by B P 8 6 . n

n

a v

n

3

n

0

3

Table V . Calculated and Experimental M -

15

Table V I . Calculated and Experimental H and M-alkyl Bond Energies** for Transition M - H and M - C H 3 Bond Energies for f-Block Elements D(M-X)- D(M-X)- Exp. Bond D(M-R) NR REL Molecule BP86 Exp. 317.7 -335 a ih-H 125. 8 3 CI3TI1-CH3 149.6 333.6 -335 (CO) Mn-H 288.13 284.2±4 CI3U-H 43.9 293.0 319.4 c

5

(CO)5Mn-CH

207.93

192±11

(CO)5Mn-CF3 (CO) Mn-C(0)CH

223.96

203±6

3

188.75

185±8

(CO) Co-H

283.11

280.1±4

(CO)4Co-CH3

197.63

5

4

a

Ref 14.

b

3

c

Energies in kJ/mol- Ref. 12f

CI3U-CH3 Cl Hf-H ChHf-CH3 3

70.2 310.6 323.5

301.8 318.1 331.9

302.6

a

Ref. llc,e. D(M-X)-NR are non relativistic bond energies whereas D(M-X)-REL includes relativistic effects according to the method of Ref. 11c. Ref. 12e. KJ/mol b

c

Compounds of f-block Elements. The final type of a - b o n d discussed here involves hydrogen or methyl bound to the f-block elements thorium and uranium ( l l c , e ) . It follows from Table V I that relativistic effects considerably strengthen the bonds to thorium and uranium. In the non-relativistic case the 5f-orbitals of lower n-quantum number are of lower energy than the 6d shell, and primarily involved i n the M - H and M - C H 3 ( M = T h , U ) bonds. Since the 5f orbitals are rather contracted the 5f overlaps with the ISH or G C H 3 orbitals are small and the M - H and M - C H 3 bonds ( M =Th,U) weak. The inclusion of relativity w i l l again reduce the kinetic energy of the s-type orbitals (including 7s) and contract their radial extent. This w i l l reduce the effective nuclear charge seen by 5 f and 6d and increase their energy (15) This indirect relativistic destabilization is largest for 5 f and 6d now becomes the predominant valence orbital. The 6d has better overlaps with ISH or GCH3- Thus, the calculated M H and M - C H 3 ( M = T h , U ) bonds become stronger i n the relativistic c a s e ( l l e ) and quite similar to the experimental estimates. W e see further by comparison that the analogous H f - H and H f - C H 3 bonds are only influenced little by relativity. Hafnium binds primarily through 5d with and without relativity. The M - C H 3 and M - H bonds are quite comparable i n strength for the f-block elements and the 5d element hafnium, with the f-elements forming bonds that are ~ 25 kJ/mol stronger.


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The Double Bond We shall now turn to a discussion of double bonds. To this end we shall make reference to the extended transition state method (2a, 16) (ETS) by which it is possible to break up the total bond energy A E A B into stabilising contributions from a,7c,8bonds, 8b-d, as well as repulsive 3- and 4-electron orbital destabilizing interactions of the type we have encountered in the inter-atomic Pauli core repulsion, 2,5.


o-bonds

7i-bonds

4-electron 2-orbital Destabilization

8a

8e

Energy Decomposition scheme Double Bonds Between Main Group Elements. We shall begin our discussion of periodic trends in double bonds by considering calculated and experimental M=M bond energies in the homologous series of ethylene type group-14 compounds m2h4, 9, (M=C,Si,Ge,Sn, and Pb), Table VII. We note first of all that BP86 is in as good agreement with experiment as the high level G l and G2 ab initio methods, Table V H In general, the M=M bond is seen to decrease in strength as we decend the group towards lead. It has previously been suggested that this trend is due to a weakening of the rc-bond. In fact, our ETS analysis as well as other studies (16) have shown that the relative importance of the 7c-bond increases compared to that of the a-component towards the heavier congeners. Instead, it is the inter-atomic Pauli core repulsions, 10, that is responsible for the weak bond in the higher homologues. That is, the two group-14 elements can not come close enough together for the heavier group-14 elements to form strong a-bonds without encountering an extensive inter-atomic Pauli core repulsions. As a result, a much longer equilibrium distance is adopted than what would give the optimal a-interaction (16). Table VII. Calculated and Experimental Bond Energies > in M 2 H 4 (M=C,Si,Ge,Sn, and Pb) a

a

b

Bond

nm

ab initio

C=C Si=Si Ge=Ge Sn=Sn Pb=Pb

w 250 180 121 42

735 246 154 119

(Gl) (G2) (HF) (HF)

Hxp 2.78 2.36 2.09 2.10 1.41

71*> 265

Inter-atomic Pauli core repulsion

54

b

Energies in kJ/Mol. Ref. 17 (n-l)p

G

(n-l)p

10


a

COMPUTATIONAL T H E R M O C H E M I S T R Y


376

Ligand to Metal Double Bonds. The Cr(CO)5 fragment is isolobal with M H 2 (M=C,Si,Ge,Sn,Pb). It can as a result replace one of the M H 2 units in the M 2 H 4 molecules, 9, to form the carbene type series, 11, C r ( C O ) 5 M H 2 (M=C,Si,Ge,Sn,Pb). Table VIII displays the calculated (CO)5Cr=MH2 bond energies decomposed into contributions from the steric (8a) interaction, A E ° , and the a- and n -bonding interactions, AEo and A E ^ , respectively. It is clear that the jr-bonding contribution, AEjt, is trend setting. It is largest for (CO)5Cr=CH2 with the strongest C r = M H 2 bond since in that case the energy gap between the occupied ;c(Cr(CO)5) and vacant TC(MH2) orbitals involved in the 7C-bond is smallest, 12. A small gap gives the strongest interaction since density goes from the lower occupied orbital ;c(Cr(CO)5) to the higher rc(MH2) orbital. There is a further decline in the (CO)5Cr=MH2 bond strength in going from Si to Sn as the TC(MH2) orbital increases slightly in energy.

9^

K

-

(EH ) 2

AER. Ref 2 2 ^ Relativistic contribution b

c

a

7r

0


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It follows that metal-metal bonds of 3d-elements are weak (60-100 kJ/mol), and the reason is again the destabilizing influence of the 3s,3p core-like orbitals on the metals. We find for a pair of homologous binuclear complexes involving 4d and 5d elements, that the 5d element invariably has the stronger metal-metal bond. The 5d elements have weaker a- and ^-interactions, A E and A E , but also a less repulsive steric interaction, A E ° , all because of a longer M - M bond compared to the 4dhomologues. The reduction in the steric interaction , AE°, is the prevailing factor in making the M - M bonds of the heavier homologues the strongest. We note that the ainteraction A E 5 is weak. It adds very little to the M - M bond strength


0

n

Table XVII. Optimized Metal-Metal Bond Distances and Decomposition of Calculated a

b

c

Bonding Energies > » for M2Cl4(PH3>4 (M=Mn,Tc,Re)

M C1 (PH )4

R(M-M) Calc.

Mn Cl4(PH )4 Tc Cl (PH ) Re Cl4(PH )4

1.92 2.17 2.29

2

4

3

2

2

2

a

3

4

3

3

4

D(M-M) calculated AE° A E -650 291 -614 355 -427 328

0

AE* 451 591 520

AE5

AE

7 5 9

11

d R

AE 99 337 441

kJ/mol. ÊÂEP+AEcpi-AE^+AEo+AER.CRef 22^ Relativistic contribution

Concluding Remarks. We have analysed some of the factors responsible for periodic trends in bond energies involving main group elements and transition metals. We have shown that the Pauli exclusion principle is responsible for many of the observed trends through intra- and inter- atomic core repulsion. Relativistic effects are also important as valence electrons near the nucleus can obtain high instantaneous velocities. Relativistic effects will reduce intra- and inter- atomic core repulsion and thus strengthen the chemical bond. Acknowledgment. The work presented here was supported by Natural Sciencess and Engineering Research Council of Canada (NSERC) as well as the donors of the Petroleum Research Fund, administered by the American Chemical Society. Thanks also to Drs. Elzbieta Folga, Jian L i , Heiko Jacobsen, Liqun Deng, Nicole Sandblom and Matthias Bickelhaupt for their contributions. Literature Cited. 1.

2.

3.

4. 5.

(a) Ziegler, T. Chem. Rev. 1991, 91, 651. (b) Parr, R. G.; Yang, W. Energy Density Functional Theory of Atoms and Molecules ,Oxford University, New York, 1989. (c) Baerends,E.J. J.Phys.Chem. 1997, in press. (a)Ziegler,T. "A General Energy Decomposition Scheme for the Study of MetalLigand Interactions in Complexes, Clusters and Solids", NATO-ASI Series ; D. Salahub (editor), 1992, C 378, 367. (b) Ziegler,T.; Tschinke,V. "Density Functional Calculations on Transition Metal Complexes", T.J.Marks (Ed.), ACS Symposium Series 428, 1990, 279 (c) Frenking, G.; Antes,I.;Böhme,M.;Dapprich,S.;Ehlers,A.W.; Jonas,V.;Neuhaus,A.; Stegmann,M.O.R.; VeldkampA.; Vyboishchikov, S.F., in Reviews in Computational Chemistry, Vol. 8, Lipkowitz, K.B. and Boyd, D.B. (Eds), VCH, New York, (1996) (a) L.Deng, V.Branchadell; T.Ziegler, J.Am.Chem.Soc. 1994,116, 10645 (b)CRC Hand book of Chemistry and Physics , edited by Lide, D.R. 74th Edition, 1993-1994. (a) Perdew, J. P. Phys. Rev. 1986, B13, 8822. (b) Becke, A. D. Phys. Rev. 1988, A38, 3098. Hehre, W.J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. In Ab Initio Molecular Orbital Theory ; Wiley-Interscience: New York, 1986. In Computational Thermochemistry; Irikura, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1905.

382

6. 7.


8. 9. 10.

11.

12

13 14 15 16 17 18 19 20 21 22

23 24

25 26 27


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4

4

2

2

4

than π ·δ , see Ref. 22d. In Computational Thermochemistry; Irikura, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1905.

Periodic Trends in Bond Energies: A Density Functional Study - ACS

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