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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.
The derivative formulation is perhaps the easiest to understand. In this
case the energy is expanded in a Taylor series in the perturbation
. . 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.
10.1.1 External Electric Field
The interaction of an electronic charge distribution p(r) with an
electric potential V(r) gives an energy correction.
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
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
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
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