Helsinki-1     

Fundamental Aspects of Chemical Bonding 

Gernot Frenking 

Fachbereich Chemie, Philipps-Universität Marburg 

   

Zusammenfassung  Das Kräftespiel zwischen neutralen Atomen zeigt eine charakteristische

quantenmechanische Mehrdeutigkeit. Diese Mehrdeutigkeit scheint geeignet zu sein, die verschiedenen

Verhaltungsweisen zu umfassen, welche die Erfahrung liefert: Bei Wasserstoff z. B. die Möglichkeit einer

homöopolaren Bindung, bzw. elastischer Reflexion, bei den Edelgasen dagegen nur die letztere — und zwar

dies bereits als Effekte erster Näherung von ungefähr der richtigen Größe. Bei der Auswahl und Diskussion

der verschiedenen Verhaltungsweisen bewährt sich das Pauliprinzip auch hier, in Anwendung auf Systeme

von mehreren Atomen.

 

Vorgetragen auf der Gauvereinstagung der Deutschen Physikalischen Gesellschaft Freiburg i. Br., 12. Juni

1927.

 

Herrn Prof. Schrödinger möchten wir von Herzen danken für das liebenswürdige, fördernde Interesse, mit

welchem er unsere Arbeit begleitet hat. Dem International Education Board danken wir, daß er es uns

ermöglicht hat, hier in Zürich zu arbeiten.

 

W.

Chemical bonding : classical approach

q

q(H2) = q(Ha) + q(Hb) E(H2) = E[q(H2)] = E11

Chemical bonding : quantum chemical approach

q(H2) = [Ψ]2 Ψ(H2) = c1 Ψ(Ha) ± c2 Ψ(Hb) [Ψ(H2)]

2 = [c1 Ψ(Ha) ± c2 Ψ(Hb)]2 = [c1 Ψ(Ha)]

2 + [c1 Ψ(Ha)]2 ± 2[c1 Ψ(Ha)c2 Ψ(Hb)]

q(Ha) q(Hb) Resonance

E(H2)α,β = E[q(H2)] ± E[Ψ(Ha) Ψ(Hb)] = E11 ± E[Ψ(Ha) Ψ(Hb)]

A + B A-B

(A) + (B) [(A) + (B)](rA-B)

Eelstat

(A,B) =NÂ(A,B)

EPauli

NÂ

(A,B) (A-B)

EOrb

1.

2.

3.

1. + 2. + 3. = Eint

Eint + Eprep = E(BDE)

Three Steps:

Energy Decomposition Analysis (EDA) Extended Transition State Method (ETS)K. Morokuma, J. Chem. Phys. 1971, 55, 1236 T. Ziegler, A. Rauk, Theor. Chim Acta 1977, 46, 1

Nature of the Chemical Bond in H-H Energy partitioning analysis of the H-H bond. Energy values are given

in kcal/mol. Bond lengths are given in Å. Experimental values in

parentheses.

Variable H2 Eint -112.9 EPauli 0.0 EElstat +5.8 EOrb -118.7 (100%) E -118.7 (100%) E 0.0 H-H bond length 0.745 (0.741)

De -112.9 D0 -106.3 (-103.3)

A. Krapp, F. M. Bickelhaupt, G. Frenking, Chem. Eur. J. 2006, 12, 9196.

Nature of the Chemical Bond in N2 Energy partitioning analysis of the N-N bond. Energy values are given

in kcal/mol. Bond lengths are given in Å. Experimental values in

parentheses.

Variable N2 Eint -232.2 EPauli 791.7 EElstat -308.5 (30.1%) EOrb -715.4 (69.9%) E -470.0 (65.7%) E -245.4 (34.3%) Overlap 1.59 Overlap 0.74 E-E bond length 1.105 (1.09768)

De -232.2 Do -228.8 (-225.0)

EE

( a )

EE

( b )

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

Overlap integral Sij between Fragment orbitals i and j

for N2

Sij

Deviation from re in Å

ss sp(ps) pp pp(p'p')

sum sum

-0.5 0.0 0.5 1.0

-1000

-800

-600

-400

-200

0

200

400

600

800

1000

EDA for N2

en

erg

y in

kca

l/m

ol

Deviation from re in Å

Eint

EPauli

EElstat

EOrb

E

E

A. Kovacs, C. Esterhuysen. G. Frenking, Chem. Eur. J. 2005, 11, 1813

A. Krapp, M. F. Bickelhaupt, G. Frenking, Chem. Eur. J. 2006, 12, 9196.

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

Overlap integral Sij between Fragment orbitals i and j

for O2

Sij

Deviation from re in Å

ss sp(ps) pp pp(p'p') sum sum

-0.5 0.0 0.5 1.0-1000

-800

-600

-400

-200

0

200

400

600

800

1000

EDA for O2

en

erg

y in

kca

l/m

ol

Deviation from re in Å

Eint

EPauli

EElstat

EOrb

E E

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

Overlap integral Sij between Fragment orbitals i and j

for F2

Sij

Deviation from re in Å

ss sp(ps)

pp pp(p'p') sum sum

-0.5 0.0 0.5 1.0

-1000

-800

-600

-400

-200

0

200

400

600

800

1000

EDA for F2

en

erg

y in

kca

l/m

ol

Deviation from re in Å

Eint

EPauli

EElstat

EOrb

E

E

1

1

Li (Na) Li (Na)

1 g

+

1

Be (Mg) Be (Mg)

1

1

1

1 g

+

1

11

1

1 1

1 1

1 1

N (P) N (P)

1 g

+

1

11

1

1

1

B (Al) B (Al)

3 g

- 1

11

1

1

1 1

O (S) O (S)

1

1

1

11

3 g

-

1

11

1

1 1

1 1

C (Si) C (Si)

1 g

+ 1

11

1

1 1

F (Cl) F (Cl)

1

1

1

11

1

1

1

1 g

+

1

11

1

1

1

1

Si Si

1

3 g

-

Table 1: Energy partitioning analysis of the first row dimers E2 (E =Li-F) in C2v at BP86/TZ2P (ZORA); energies in kcal/mol, distances E-E in Å

E Li Be B C N O F

el. State 1g+ 1

g+ 3

g - 1

g+ 1

g+ 3

g- 1

g+

Eint -20.71 -7.86 -74.67 -140.79 -240.23 -158.99 -52.87 EPauli 1.76 41.62 135.00 252.20 802.37 462.39 146.07 EElstat

a -8.30 (36.9%) -17.87 (36.1%) -33.14 (15.8%) -3.22 (0.8%) -312.85 (30.0%) -159.74 (25.7%) -41.20 (20.7%) EOrb

a -14.17 (63.1%) -31.62 (63.9%) -176.53 (84.2%) -389.77 (99.2%) -729.76 (70.0%) -461.65 (74.3%) -157.75 (79.3%)

A1b (σ) -14.17 -31.62 -104.50 (59.2%) -201.74 (51.8%) -478.81 (65.6%) -327.69 (71.0%) -151.49 (96.0%)

A2

b (δ) 0.00 0.00 0.00 0.00 0.00 0.00 0.00

B1 b (π) 0.00 0.00 -36.02 (20.4%) -94.02 (24.1%) -125.47 (17.2%) -63.19 (13.7%) -3.13 (2.0%)

B2

b (π) 0.00 0.00 -36.02 (20.4%) -94.02 (24.1%) -125.47 (17.2%) -70.77 (15.3%) -3.13 (2.0%) Ecorr.

e 0.29 0.00 1.92 3.27 4.15 5.30 2.72 Eprep 0.00 0.00 0.00 0.00 0.00 16.64 0.00 De

c 20.42 (24.62)

7.86 (2.28;

2.70[calc.]) d

72.75 (71.15)

137.52 (145.86)

236.08 (228.43)

137.05 (120.23)

50.15 (38.25)

E-Ec 2.731

(2.673) 2.442

(2.45) d 1.617

(1.590) 1.253

(1.243) 1.102

(1.098) 1.224

(1.208) 1.420

(1.412) a Values in parenthesis give the percentage contribution to the total attractive interactions Eelstat+Eorb b Values in parenthesis give the percentage contribution to the total orbital interactions Eorb c Experimental values in parenthesis from Ref. 11a unless otherwise specified.

d Experimental value for De and E-E V.E. Bondybey, Chem. Phys. Lett. 1984, 109, 436; calculated value for De J. M. L. Martin, Chem. Phys. Lett. 1999,

303, 399. eCorrection for the spin polarization

N2 P2 As2 Sb2 Bi2

Eint -232.2 -109.2 -80.6 -54.4 -48.4

EPauli 791.7 299.3 247.9 182.3 168.1

Eelstat -308.5 (30.1%) -175.8 (43.0%) -160.5 (48.9%) -131.5 (55.6%) -126.3 (58.3%)

Eorb -715.4 (69.9%) -232.7 (57.0%) -168.0 (51.1%) -105.2 (44.4%) -90.3 (41.7%)

E -470.0 (65.7%) -140.1 (60.0%) -105.1 (62.6%) -69.9 (66.4%) -61.2 (67.8%)

E -245.4 (34.3%) -92.6 (40.0%) -62.9 (37.4%) -35.3 (35.6%) -29.1 (32.2%)

R(E-E) 1.105 (1.0977) 1.935 (1.8931) 2.161 (2.103) 2.579 (2.48) 2.728 (2.660)

De -232.2 -109.2 -80.6 -54.4 -48.4

Do -228.8 (-225.0) -108.1 (-116.1) -80.0 (-91.3) -54.0 (71.3) -48.1 (-47.0)

Table 2. Energy partitioning analysis of the N-N, C-O and B-F

bonds. Energy values are given in kcal/mol. Bond lengths are

given in Å. Experimental values are given in parentheses.

N2 CO BF

Eint -232.2 -258.4 -180.8

EPauli 791.7 575.8 476.1

EElstat -308.5 (30.1%) -240.0 (28.8%) -210.5 (32.0%)

EOrb -715.4 (69.9%) -594.2 (71.2%) -446.4 (68.0%)

E -470.0 (65.7%) -301.7 (50.8%) -396.4 (88.8%)

E -245.4 (34.3%) -292.5 (49.2%) -50.0 (11.2%)

bond length 1.105 (1.09768) 1.144 (1.128) 1.285 (1.262)

De -232.2 -258.4 -180.8

D0 -228.8 (-225.0) -255.4 (-255.7±1) -178.9 (179.9±3)