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)