Chapter 4
Introduction to Quantum Mechanics in
Computational Chemistry
It is by logic that we prove. but by intuition that we discover.
J. H. Poincar, ca. 1900.
4.1 PERSPECTIVE
Chapter 1 outlined the tools that computational chemists have at their disposa
108 Computational Chemistry
Now, it can be proved that if and only if A is a symmetric matrix (or more generally, if
we are using complex numbers, a Hermitian matrix see symmetric matrices, above),
then P is orthogonal (or more generally, unitary see orth
94 Computational Chemistry
where (b is the amplitude of the particle/wave at a distance x from some chosen origin.
The 1D Schrodinger equation is easily elevated to 3D status by replacing the 1D operator
dz/dx2 by its 3D analogue
32 32 32
_. _ _ = V2 4.28
114 Computational Chemistry
the system of equations (4.49) we note that in the first equation (row 1), the coefficient
of C] has the subscripts l 1 (row 1, column 1) and the coefficient of c2 has the subscripts
12 (row 1, column 2), while in the second eq
82 Computational Chemistry
Schrodinger equation, and then the birth of quantum chemistry with (at least as far
as molecules of reasonable size goes) the application of the Schrodinger equation to
chemistry by Hiickel. This simple Hiickel method is current
Molecular Mechanics 63
cfw_ .
| (CH2)n _. A
( : (CH2)H
*J \J
2 3
N2
0 H o: i H 2 :EH
5 6 4
Figure 3.12. Some molecules (1, 2, 4) which have been synthesized with the aid of MM.
Enediynes like 2 (Fig. 3.12) are able to undergo cyclization to a phenyltype d
100 Computational Chemistry
The 0 2p electron density
(the square of the waveluncon) C than like this: 0 The wavelunctlon
looks more like this: 0 O itself looks like 1:
Hence pp overlap looks like this: rather than like this:
00 00
C32) 00
Figure 4.8. The
Molecular Mechanics 61
One may wish a more precise approximation to the transition state geometry than
is represented by an intermediate or a compound somewhat resembling the transition
state. This can sometimes be achieved by optimizing to a minimum subj
112 Computational Chemistry
. Energy
Two sAOs Two Sigma-type M08
or m cm 022% 9 _ G AnhbondlngM
M02 I92
A node \cfw_AOs change sign here)
oh is c MC\ 99 Bond
/ n 1 all: mg M0
Cosmsisni ol basis tween 1 Coalclent of basis funclion 2 M01 5"
innit)I innol
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Input structure with this symmetry will be
Energy optimized to the transition state
-. - Input sluctures not of 03., symmetry will
_ be optimized to the minimum-energy
.-" conformation
Dlhedral angle
Figure 3.9. Optimi
78
[35]
Computational Chemistry
S. Grime, J. Am. Chem. Soc., 1996, 118, 1529; (h) V. Balaji, J. Michl, Pure Appl. Chem.,
1988, 60, 189; (i) K. B. Wiberg, J. W. Ochterski, J. Comp. Chem., 1997, I8, 108.
J. l. Seeman, Chem. Rev., 1983, 83, 83.
EASIER QUESTI
Introduction to Quantum Mechanics 107
be calculated from vector algebra: the dot product (scalar product) is
V1 - V2 = lVIHV2l0089
where |v| ("mod V) is the absolute value of the vector, i.e. its length:
2 2 l 2 2 2 2 l 2
IVI = (U, + vy) / (or (vx + vy +
Molecular Mechanics 57
(structure 1) to the quite unnatural length of 1.600 A (structure 2) will lower the poten
tial energy by 67 kJ mol" 1 , and indicates that the drop in energy is due very largely to the
relief of nonbonded interactions. A calculation
Introduction to Quantum Mechanics 91
4.2.6 The wave mechanical atom and the Schrodinger equation
The Bohr approach works well for hydrogen-like atoms, atoms with one electron:
hydrogen, singly-ionized helium, doubly-ionized lithium, etc. However, it showe
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Late nineteenth century physics, classical physics at its zenith, predicted that the ux
density emitted by a black body should rise without limit as the wavelength decreases.
This is because classical physics held that radiation
Introduction to Quantum Mechanics 101
pl bond
sigma bond
Approximately
Figure 4.10. The model of a C/C double bond as a 0/7: bond is at bottom really equivalent to the
spS/sp5 + sp5 /sp5 model: both result in the same electron distribution, which is t
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Comparing Eq. (4.66) with Eq. (4.60), we see that we have obtained the matrices
we want: the coefficients matrix C and the MO energy levels matrix a. The columns of
C are eigenvectors, and the diagonal elements of e are eigenva
104 Computational Chemistry
To get an idea of why matrices are useful in dealing with systems of linear equations.
let us go back to our system of equations
(1) aux + my = Cl
(2) azrx +1122) = 02
Provided certain conditions are met this can be solved for
110 Computational Chemistry
a proof, a method that gives useful results and which can be extended to more powerful
methods with the retention of many useful concepts from the simple approach.
The Schrodinger equation (section 4.2.6, Eq. (4.29)
8n2m
v2+ h,
90 Computational Chemistry
to small-radius orbits. The fading of quantum-mechanical equations into their classi-
cal analogues as macroscopic conditions are approached is called the correspondence
principle [11].
Using the postulate of Eq. (4.7) and class
Introduction to Quantum Mechanics 99
Another example illustrates a situation somewhat similar to that we saw with methane,
and what was until fairly recently a serious controversy: the best way to represent the
carbon/carbon double bond [28]. The currentl
Molecular Mechanics 69
(1) Characterizing a species. This is not often done with MM, because MM is
used mostly to create input structures for other kinds of calculations, and to study
known (often biological) molecules. Nevertheless MM can yield informati
88 Cotilputational Chemistry
Brownian motion that the reality of atoms was at last accepted by such eminent holdouts
as Boltzmanns opponent Ostwald.ll
The atom has an internal structure; it is thus not atomic in the Greek sense and is
more than the mere r
Introduction to Quantum Mechanics 109
1, +1, 1. It is also possible to start at, say element (2,1), the number 1, and move
across the second row (, +, , +), or to start at element (1.2) and go down the column
( , + , , + ) , etc. One would likely choose t
106 Computational Chemistry
(6) The transpose (AT or A) of a matrix A is made by exchanging rows and columns.
Examples:
_ 2 3 T_ 2 4
HA (4 7), thenA (3 7)
2 l
IrA=G ; g), thenAT= 1 7
6 2
Note that the transpose arises from twisting the matrix around to in
96 Computational Chemistry
N i-e-
I a. 5
(311-13
La.
06%
Figure 4.4. The SHM is used mainly for planar arrays of 7: systems.
benzene is aromatic but cyclobutadiene is not [17]. With the application of computers
to quantum chemistry the MO method almost ec
Introduction to Quantum Mechanics 105
(6) the transpose of another matrix.
(7) orthogonal matrices.
(1) The zero matrix or null matrix. 0. is any matrix with all its elements zero.
Examples:
0 0 0 0 0
(00)(000)(0000)
Clearly, multiplication by the zero ma
7O Computational Chemistry
ACETONE
30 0
301 I
an
2990
40 so
20
MM3 11'?!
0 Expedmenl Int:
4000 soon 2000 1000 D
100
100
Figure 3.14. Experimental (gas phase) and MM (MM3) and MP2(FC)!6-3lG* calculated infra
red spectra of acetone. The MM3 spectrum
Introduction to Quantum Mechanics 111
The integration variable dv indicates integration with respect to spatial coordinates (x, y,
z in a Cartesian coordinate system), and integration over all of space is implied, since
that is the domain of an electron i
Introduction to Quantum Mechanics 119
=8% 2 (1
G
lb)
/2\ a '9 o
1 s :25
-=+or- or
(a)
a t9
,6 at
D as
2 p o
(a)
Ian/I
x/OIE
:3
0
13
at
Figure 4.14. Some conjugated molecules, their p orbital arrays, simplied representations of the
molecules, and their sim
Introduction to Quantum Mechanics 95
mm = mg there is a mathematical function obtained by combining the appropriate
f (r), f (9) and f (05):
Mr, 9, <15; '1', 1', min) = f(r)f(0)f() (4.32)
The function 1b(r, 6, ) (clearly 30 could also be expressed in Cart