# Introduction to Computational Chemistry - Jensen Fr.

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fields, may either be time independent, leading to static properties, or

time dependent, leading to dynamic properties. Time-dependent fields are

usually associated with electromagnetic radiation characterized by a

frequency. Static properties may be considered as the limiting case of

dynamic properties when the frequency goes to zero. We will only consider

the static case here, and again concentrate on properties of a single

molecule for a fixed geometry. A direct comparison with (gas phase)

experimental macroscopic quantities may be done by proper averaging over

for example vibrational and rotational states. We will furthermore

concentrate on the electronic contribution to properties, the

corresponding nuclear contribution (if present) is normally trivial to

calculate as it is independent of the wave function.1

There are three main methods for calculating the effect of a

perturbation: derivative techniques, perturbation theory and propagator

methods. The former two are closely related while propagator methods are

somewhat different, and will be discussed separately.

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MOLECULAR PROPERTIES

The derivative formulation is perhaps the easiest to understand. In this

case the energy is expanded in a Taylor series in the perturbation

strength E.

. . 1 aaA e 2 1 d3E ,

A(O)-A(0) + ?\ + --\ +^x +... (10.1)

The nth-order property is the nth-order derivative of the energy, aiA/aA"

(the factor 1 jn\ may or may not be included in the property). Note that

the perturbation is usually a vector, and the first derivative is

therefore also a vector, the second derivative a matrix, the third

derivative a (third-order) tensor etc.

10.1 Examples

10.1.1 External Electric Field

The interaction of an electronic charge distribution p(r) with an

electric potential V(r) gives an energy correction.

E =

p(r)V(r)dr (?.2)

Since the electric field (F = -dV/dr) normally is fairly uniform at the

molecular level, it is useful to write ? as a multipole expansion.

E - qV - jiF - |QF' - ... (10.3)

Here q is the net charge (monopole), o is the (electric) dipole moment, Q

is the quadrupole moment, and F and F' are the field and field gradient

(<9F/<9r), respectively. The dipole moment and electric field are

vectors, and the jiF term should be interpreted as the dot product (o, ¦

F = \ixF x + jiyFy + pzFz). The quadrupole moment and field gradient are

3 x 3 matrices, and QF' is the sum of all product terms. For an external

field it is rarely necessary to go beyond the quadrupole term, but for

molecular interactions the octupole moment may also be important (it is

for example the first nonvanishing moment for spherical molecules like

CH4).

In the absence of an external field, the unperturbed dipole and

quadrupole moments may be calculated from the electronic wave function as

simple expectation values.

OIO|A|O) (10 4)

Q = (O|aa*|O) ^

The superscript t denotes a transposition of the r-vector, i.e.

converting it from a column to a row vector. The rrl notation for the

quadrupole moment therefore indicates a 3 x 3 matrix containing the

products of the x-, y- and z-coordinates, e.g. the Qxy component is

calculated as the expectation value of xy.

The presence of a field influences the wave function, and leads to

induced dipole, quadrupole etc. moments. For the dipole moment this may

be written as

H = H0 + aF + ipF2+iyF3 + ... (10.5)

where o0 is the permanent dipole moment, a is the (dipole)

polarizability, p is the (first) hyperpolarizability, o is the second

hyperpolarizability etc. The quadrupole moment

10.1 EXAMPLES

237

may similarly be expanded in the field by means of a quadrupole

polarizability, hyperpolarizability etc.

For a homogeneous field (i.e. the field gradient and higher derivatives

are zero), the total energy of a neutral molecule may be written as a

Taylor expansion.

AII = E(0) +|§F + I2!f +1 "V + + . •. (19.6)

v ; ' dF 23F2 6<9F3 24<9F

According to eq. (10.3) we also have E = - o • F, where o is given by the

expression in eq. (10.5). Carrying out the differentiation in eq. (10.6)

shows that the first derivative is the (permanent) dipole moment o0, the

second derivative is the polarizability a, the third derivative is the

hyperpolarizability p etc.

?(F) = ?(0) - fi0F - i"F2 - iPF3 - iYF4 - ... (10.7)

Note that the constant factor in front of the higher-order terms differs

between eqs. (10.3) and (10.5)/(10.7).

10.1.2 External Magnetic Field

The interaction with a magnetic field may similarly be written in term of

magnetic dipole, quadrupole etc. moments (there is no magnetic monopole,

corresponding to electric charge). Since the magnetic interaction is

substantially smaller in magnitude than the electric, only the dipole

term is normally considered.

E = -mB - ... (10.8)

The dipole moment m depends on the total angular momentum, which may be

written in terms of the orbital angular moment operator L and the total

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