Appendix A
Calculation of absorption spectra by
time-dependent methods
The following derivations are presented here for reasons of self-consistency of this
thesis. Original derivations have been given in the literature, using, in part, different notations, e.g. Heller [65].
A.1 Derivation of a microscopic expression for the
absorption cross section using time dependent
perturbation theory
The intensity I (energy crossing through a unit area per unit time) of a light beam
passing in z-direction through a gas of molecules which absorbs electromagnetic
energy and converts it into internal (for example vibrational) energy decreases according to Beer’s law [41],
(A.1)
"!#
In equation (A.1), is the density (number of molecules per volume) and
is
the molecule-characteristic and frequency- (but not intensity-) dependent absorption cross section.
128
Appendix A: Time dependent perturbation theory
129
For the derivation of the theoretical absorption cross section by time-dependent
perturbation theory we start with the one-dimensional time-dependent Schrödinger
equation in the Born-Oppenheimer approximation:
$%'& ,
0 -/. 1243+5 (768
&)(+*
F
:QP
F
9;: =% < <
&
0 1D243 76 E O,
0 -. 1235 (76
>+? 3 2 3A
< C
& G4H @ B
I *
JLKNM
U
1 (76SR+TU ? 0 U D1 243 6 R O, -. D1 243+5 (76 5
*
G4H
I
JLV)WYZ\X []_^
(A.2)
In equation (A.2) the interaction with the electric field is described within the electric dipole approximation. The real dimensionless parameter ` is much smaller
b
than one to assure a small perturbation by the electric field a 1 (76 in comparison
with the unperturbed Hamiltonian c\d .
In the case of the time-independent Hamiltonian c\d , the formal solution of the
time-dependent Schrödinger equation is
0 1 (768feg - K h[ ] g ] M ^_ikj 0 1 ( d 6ml
*
*
,O-.
The eigenfunctions of the operator c\d , n ,po , with
,O-.
,O-.
cqa drn , o 8fs , o n , o
(A.3)
(A.4)
can be used to expand the time-dependent vibrational nuclear wavefunction
,O0 -. 1
t
t
(76 :
*
*
,O0 -. 1
(767uY8 Twv
v
oLx
where
v
v
o 1 (768zy n O, -o .
x
v
o 1 (76 n O, -o . u 5
(A.5)
t
*
,O0
-. 1
(767ul
(A.6)
Expansion (A.5) holds for a discrete
eigenbasis:
t and complete
t
T
,ko
,O-.
O, -.
n p, o {u y n k, o 8}| l
(A.7)
Only bound-bound transitions are considered in this section. A generalization to
t
bound-free transitions is given by Schinke [41]. Multiplication on the left with
,-/.
y n ,po yields (index 243 denotes an integration over 23 t=in
‚ coordinatet space):
v 
$O% & , o 1
1 (764y n O,, o-.
(768~s , o R , o 1 (76 @ ` R TU€T v
&)( x
x
x
:ƒP
v 
1 (76 ? 0 U…„ n ,-/. 7u †D‡l (A.8)
Appendix A: Time dependent perturbation theory
130
An electronic two-level system is assumed where the initial (ˆ ) state (‰ = ˆ ) (e.g.
the electronic ground state) is coupled with only one final ( Š ) state ( ‹ = Š ) (e.g. an
electronic excited state) by an electric field. The time-dependent equation for the
final state Š reads then as:
(A.9)
ˆOŒ\
ž Ž7“4£¥¤ O ¦§
Ž7“•”f–
Ž7“š™œ›
7Ž “
¤ žO ¦ § °7±D²+³
)ŽrO‘=’
p‘˜—rO‘=’
—+Ÿž¡ ¢ ’
p ‘©¨=ª«ƒ¬ ’ — ­ ® ¦Ÿ¯ ¨
where
is the transition dipole moment that connects the initial (ˆ ) with the final
­ ® ¦ In the absence of the electric field (
( Š ) state.
) the solution of the last equation
›´”~µ
is very simple:
rO‘=’
Ž7“•”~¶
ŽO·m“¸¹ ¦º¡» ‘+¼½ ¹ ½¿¾ÀÂÁkÃÄ
p‘=’
(A.10)
With this solution equation (A.5) becomes
Ž7“ ° ”

¨Å ® ’
O‘
ŽO·…“7¸¹ ¦º» ‘+¼½ ¹ ½ ¾ _À Ákà ¤  ¦/§ °
rO‘’
¨ O ‘
(A.11)
which is a coherent superposition of eigenstates (i.e. a wavepacket). Substituting
(A.11) into (A.3), one obtains

O‘
Ç
ŽO·…“¸ ¹ ¦º » ‘ ¼½ ¹ ½ ¾ ÀÂÁkà ¤ O¦§ ° ”f¸ ¹ ¦ Æ ¼h½ ¹ ½ ¾ _À ÁkÃ
mp‘’
¨ O‘
ȝ
Ç
É4Ê
Ë
È
ŽO·…“)¸ ¹ ¦ º » ‘ Ìμ½ Í ¾ ¹ ½ ¾ À_Ákà ¤ O ¦§ °
mO‘’
¨ O ‘ Ë
É4Ê
O‘
Ï
‘ ¼h½¿¾À
(A.12)
”~¸ ¹ ¦ Æ ¼½ ¹ ¿½ ¾ÀÂÁkÃ
ŽO·…“ ¤ O¦§ ° Ä
 rO‘’ ¨ O‘
p‘
Ç
can be replaced by the relative phase
This means that the propagator
¸ ¹ ¦ Æ ¼½ ¹ ½ ¾ À_ÁkÃ
factors
. Only if the initial state is an eigenstate (e.g. ž
¸¹ ¦º » ‘ ¼h½ ¾ ¹ ½ ¾ À_ÁkÃ
”
Ð

‘
for
Ñ
)
the
time
evolution
is
described
by
a
global
phase
factor,
³ Ç ”}µ
®Ó”Õ
Ò Ô ®
mO‘
, with no physical importance.
¸ ¹ ¦ Æ ¼½ ¹ ½ ¾ ÀÂÁkà ”f¸¹ ¦ºÖ ‘ ¼½ ¹ ½ ¾ ÀÂÁkÃ
If the perturbation by the electric field is small, the solution of equation (A.9)
is close to solution (A.10), i. e. the solution in the unperturbed case. The solution
for small perturbations can be predicted to be
m ‘ ’
Ž7“•”f¶
‘ ’
× Ž7“¸¹ º » ‘…½ ³
(A.13)
where in opposition to equation (A.10) the coefficient
is a function of time.
¶ Ž7“
O‘=’
Substituting (A.13) into (A.9), one obtains:
ˆŒ
× × 

ˆ Œ
¸¹ º » ‘ ½
¶ Ž7“š™ È ¶ Ž7“ É4Ê ¸ ¹ º» ‘ Ë ½
—
— )Ž O ‘=’
— O ‘=’ — )Ž {× Ú
Ì
º » ‘ § » ‘ ¼h½_À¿ØÙ » ‘4Û
Appendix A: Time dependent perturbation theory
131
ò
ò
ÞOõö
Þ
O
õ
ö
ü
ï
ï
û
à âDã7ä ï å æ4è ç é ç ë
ÞOß#÷=øùƒú âŸã7ä
õŸý ÷
ç
çþ7ÿ
çYókô
ô
ÜfÝ©ÞOß{àáÞOß=âŸã7ä7åæè4ç éê ßëíìCîðï+ñDò
(A.14)
è ç éê ß ë leads to:
ò
ò
Þ
O
õ
ö
ÞOõö ï åšè ç hé ê ß æ é ë
î
ï
ï
ï
û
D
ñ
ò
ü
àrÞ ß âDã7äÜ
à âDã7ä
ù ú âDã7ä
÷
Þß ÷ ø Q
"
õ
ý
ç
ã
ç
ç þ7ÿ
kó ô
ô
ç
Multiplying both sides of the last equation by å
(A.15)
ò ã the system is assumed to be represented by the vibrational eigenfunction
For
ÞOõö
õ (e.g. the vibrational ground state ( õ ò = 0) of the electronic ground state ( =
ç ç
0)).
ô The only non-vanishing coefficient is à ç õ ç âŸãä•Ü which is independent
of time, since there is no perturbation of the system:
ò
à
ò
õ âŸãä•ÜYà âŸãäYÜ else ç ç
ç
and àáÞ ß âŸãä•Ü
and therefore:
ò
õOâŸãä•Ü
ü
(A.16)
âDãä•Ü
ô
ÞOõö
ç
(A.17)
For ã"!# equation (A.15) may be solved by first-order perturbation theory [51].
As the perturbation is increased from zero to a finite value, the new energy and
wave function must also change continuously, and they can be written as a Taylor
î
expansion in powers of the perturbation parameter .
à%$íâDã7ä•Ü î'&Yï à
$
& âDã7ä ìœî)(Sï à
$
( âDã7ä ìœî'*ï à
$
* âŸã7ä ì
(A.18)
Ü
(A.19)
Substituting this expansion into (A.15), one obtains:
î+&ï
,
à & âDã7ä ìCî-(Sï
ã OÞ ß
ò
îðï+î'&ï ñ ò
à
ç
îðï+î)( ï ñDò
ç
à
,
à ( âDã7ä ìCî+*ï
ã OÞ ß
à * âŸã7ä ì
ã OÞ ß
ò
O
Þ
õ
ö
Þ õ/ö
ï
û
ü
ÞOß ÷ ø ùQú âDã7ä
õŸý ÷
ç
ç 7þ ÿ
kó ô
ô
ò
ò
O
Þ
õ
ö
Þõ/ö
( ï
ï
û
ü
âDã7ä
Þ ß ÷ ø ùQú âDã7ä
õŸý ÷
ç
ç 7þ ÿ
ókô
ô
& Dâ ã7ä ï
ï å è ç hé¡ê ß æ é ë ì
ç
ï åšè ç hé ê ß æ é ë ì
ç
Appendix A: Time dependent perturbation theory
56?>A@CB
.0/1.'23/45687:9;=<
>@CB
/DFEHGIKJ
GLM?NOQP
/:R S
.
Collecting terms with the same power of
>Yhji
56:VCWYX
EUGIKJ
M
IT
132
6
6
/[Z \ 9^]_ L` ]a
<cbedfff
(A.20)
gives:
klBnm8o,prq
>@CBsi
@ 7 9g <
q
G L
.+g
k
6
>Yh8iut1Bnm"o,p
. ;
q
>@CBvi
@ 7 9 ;=<
G L
q
56 >@CB
7 9g <
4A5w6
>A@CB
/?DxE
/[R S
GIKJ
G LyM N OHP
M
IT
E
56 VCWX
GIKJ
/1Z \ 9z] _ L1` ] a
..
.
>h|i}~Bm€op
.'{
q
q
>@CBsi
@ 7 ‚9 <
GL
The zeroth-order (r=0) solution for t
(A.16):
56‡†
6 >@
7
I
>A@CB
/DxE
5w6 >A@
kBvitˆ
…
M
E
/1Z \ 9^]_ L„` ]wa
ˆ
else
klBvi
GL
k
…
>@
and
IAT
klBvi
7
7
/[R S
GIKJ
GƒL M N OHP
56 VCWX
GIKJ
0 is identical with the initial conditions
…
k
f
…
(A.21)
This result permits one to write the first-order (r=1) solution in the
form:
6
o,p
q
q
@ 7
>@CBvi
IAT
6 V,WYX
Š
EHGIKJ
M
I

6 B
>@CB
DFEHG,I^J
GLM?N‰OQP
GL
where
Ž
/:R S
>=
6 i
S
G
G L I
pO
6
/1Z I^bŒ‹  _ L
I
ˆ
(A.22)
(A.23)
is the transition frequency (Bohr frequency). The first-order solution is valid for
small coupling elements or more precisely,
if
p
@’ˆ
>@CB
M
DFE
6
/[R S
GIKJ
GL M?NOQP
IAT
@
M
E
VCWX
G,I^J
(A.24)
M‘
I
where is the time duration of the perturbation. Within this approximation it is explicitly assumed that the populations of the initial and the final state do not significantly change while the light beam is switched on, i. e. (A.21) holds approximately.
If the electric field is described by a cosine-function,
Ž
Ž
P
>A@CB3i
>
P g
Z
/1“•”„–
<Kb
6
>@CB
56
7 9z ` ;=<
4 56
6
@CBvi
>
P g
/1“—”‰–
@CB—ˆ
(A.25)
6
<Kb
Appendix A: Time dependent perturbation theory
133
with ˜š™œ›ž ˜š™  and Ÿ being the unit vector in the direction of the electric field, one
obtains: ,¡r¢
£C©
¢e£„¤¦¥§?¨
¥­K®
¥ƒ§
›«ªx¬
?¯‰°H˜ ™s±[² ³
›ÃÂ
where the shorthand notation
Ÿ
›ÉªF¬
(A.27)
Ê¬
±[² ³
was introduced. Integration of (A.26) yields [41]:
¤¦¥§?¨
Â
›
ÁÖÕ
­Ì‚º1» § ¶ ¶YÍ ºÎ ½
¥ƒ¡ §š­K¶
£C©
Å
Æ
°
ÁÏ¥Ÿ § ­K¶
Ä
±sË
(A.26)
¥­K®
­K¶Q·C¸¹
­´
 ¯ °H˜ ™
±1¾—¿‰À
Ÿ„Ç
¥­K®
¥§
¥§š­^¶
Æ
Á
­‡Ìк1» § ¶ ¶
°
ÁÑ¥ Ÿ § ­K¶
Ä
£C©
­zº ½ ©’È
­‚º ½'Æ
±~Å Ä
¨AÁ
± Ÿ
¨
± Ÿ
Â
µ¬
­zº[» § ¶ ¶x¼ ½
¥ § ­K¶
­zº1» § ¶ ¶F¼ ½
¥­K®
­^¶H·,¸Y¹
­A´
º‰Î ½
Á
Ç
ÒÔÓ
(A.28)
°
ÁÑ¥§š­^¶
For
the second term of (A.28) is dominant 1 . The first term can be neglected under the condition that
ÁÑ¥§C­^¶
Ä
Ø
£’È
Õ
Áu×؊كÚYÛÝÜ
Ä
(A.29)
ÙFÚ=Û
Ț£ƒà the robeing the oscillation period of the electromagnetic field. This is called
£C©
©
tating wave approximation
(RWA) and¨ £Cimplies
that during the interval ÞÊß
many
¨
oscillations of ˜
are possible, i. e. ˜
acts as a cosine-perturbation with a cer¡
tain oscillation period. Combining conditions
(A.24) and (A.29), it follows:
¡
£
Á
³
Ä
» ­K¶yÜ
Ü
ÁÑ¥ § ­K¶
¥ƒ§š­K¶yá
¥ §
݉
Â
¥§
­K¶äã
¥ § ­K¶
°Öâ
Â
Ó
(A.30)
­K¶
This means that ¥ƒthe
energy difference â
must be much larger than the coup°€â
§%­^¶
ling element Â
. If (A.30) is fulfilled, the first-order solution and the rotating
wave approximation are valid and the population of the final state can be written
xë
å
as [50, 41]:
©
£Cì
à
¥ƒ§š­K¶,¨
£C©
¤%¥ƒ§l¨
›æ
¡ ¥§š­K¶ ê
£C©
 ç›éè
¨ŒÁÑ¥§š­K¶
ç
Â
±
ç Þ
À
Á
°
¨AÁÑ¥§š­K¶
Á
©
°
È
Å
±
(A.31)
ç
­
where the index­ denotes the dependence of the population of the final state on
xë
the initial state . Using
©
£Cì
à
¨ŒÁÑ¥§š­K¶
îï ñ
½‡í ðœ
À
ç Þ
Á
¨AÁÑ¥ § ­K¶
°
°
1
ôeõ ö
Á
©
ç
±
£
Å
›#ò Å
±
¨AÁÑ¥§š­K¶
±1ó
°
Á
©—È
(A.32)
The reverse holds for stimulated emission where ôeõFö÷‚ø)ùûú [41]. Here, only absorption,
÷zø‘üýú , is considered.
Appendix A: Time dependent perturbation theory
134
equation (A.31) reads as
ÿ
þäÿ
Ñÿ !"
$#
(A.33)
þÑÿ
is sometimes called the transition probability. According to (A.33) it rises
þäÿ%
linearly with time. The transition rate is defined as the time-derivative of
&
and therefore is time-independent:
ÿ'
(
(
þÑÿ)*
ÿ)
+
Ñÿ),!"
-#
(A.34)
Equation (A.34) yields the constant transition probability per unit time. The decrease of intensity . of a light beam passing through a gas of absorbing molecules
&
can be written as [41]:
( .
132
(0/
6
4 587
>
Ñÿ
ÿ
? @
A
CEDGFIH
;=< 59 ;
:
.
#
(A.35)
B
Inserting equation (A.34) into (A.35) leads to the microscopic expression of the
phenomenological partial absorption cross section defined in equation (A.1):
J ÿ
"
K
V D)XcH ed
4L
5 7
Ñÿ
MÏÿ
NO"
;QPSR
ÿ
ÿT
;0U WV YD XS\
H
Z [
;]R
ÿT
3^_` ;
ba
(A.36)
V
Z denotes the transition dipole moment in the direction of the
where Z d
electric field (unit vector ). The cross section defined in equation (A.36) is an
example of the Fermi’s Golden Rule. The total absorption cross section resulting
from excitation to the electronic state f is just the sum of the individual partial
cross sections of different vibrational levels g Z :
JNZ hQijh "
kml
ÿ
J ÿ%"
-#
(A.37)
If more than one excited (final) state is involved, the absorption spectrum results
from the summation over all electronic states:
JNhQ ich "
kml
Z
JNZ hQic)h M"
$#
(A.38)
A.2 Time-dependent calculation of the absorption
cross section: The autocorrelation function
The aim of this section is to show how the absorption spectrum can be calculated
using time-dependent wavepackets. The central quantity in this business is the
autocorrelation function which will be defined below.
Appendix A: Time dependent perturbation theory
135
According to (A.36) and (A.37) the total absorption cross section of a final electronic state n is given by
oNr8
pQqcs)p tuMv"wKxzyj{|
o
{|
{
{|
st uv"w}xzyj{|
st8†‡Sˆ
~ ‚8ƒ…„ v
€6
{
{|s)‰
†0ŠŒ‹3 r
)Ž c’
s ‘
†=ˆ
{|
s‰
st ‹
s)t“%”–• † — „˜ uv
v"w
(A.39)
In the first step of the derivation we use the symmetry of the Dirac’s ˜ -function,
{|
st,‹
˜ uv
™
˜ uš v s t’
› œ
A
ž6Ÿ
v"wkx
v
{|
‹ v

w$ and the possibility to write it as
{|
˜ uv¡ ‹
v
wkx
|
£ ¢
«Œ¬%­ s p Ž ž Ÿ ª ž®
¯
(A.40)
ª
~¥¤§¦©¨
Substituting this into equation (A.39), one gets
¨
oNr8
pQqcs)p tuMv"wkx
š
{
‡Sˆ
st
s‰
{
† Š ‹3 r
)Ž cs  ‘
†=ˆ
{ |s‰
„ £ ¢
› œ ~
A
°± qc²E³p
~ ‚8ƒ
€6
¢

ª
¤
¸
‡Sˆ
y {´|
µ¶-· qcpQqc²
„
›A| œ 
t
°¸ Ž ž ® ª ž t 
š
¦©¨
¨
{
|bº
­ ª s pQ¹ ®
“” •
«Œ¬
„
{ |s‰
{
† Š ‹W r
YŽ Ss  ‘
†]ˆ
st
s‰
“” • „
­ sž Ÿp ¯
(A.41)
Using expressions (A.7) and (A.12 ), equation (A.41) becomes
{
o,r
pQqcsp tuMv"wkx
ƒ¼»½*¾A¬
„ µ¶-· qcpQqc²
„
«Œ¬
ª
¤¿¦©¨
t
­
Å ÇÆ p 0
ŽYSsÀ ‘ ˆ s)t † ª YŽ S\
‡LŠI‹3 r
† ŠŒ‹3 r
s ‘
š
 š
| ›Aœ
| ›Aœ
YÁ  ® tGà ‚
 ® tÈÃ
Ž –Ä
Ä
Ž p ʐ É
s‰
{
ˆ
s‰
­ sž Ÿp ¯
stW“”• „

¨
(A.42)
The essential step for the calculation of the absorption spectrum by wavepacket
propagations is the following ansatz for the initial wavefunction in the excited state
n ;
{
s‰
† Ë rÌu–́w“x
YŽ Ss  ˆÎs)t “ ¯
†  r
(A.43)
The autocorrelation function is defined as
{
Ï
¬
r u wx
{
s ‰
s)‰ ¬
‡–Ë rÌu–́w † Ë Ì
r u w“%” • ¯
(A.44)
Finally, substituting this definition into (A.42) yields:
oNr
pQqcsp t uv"wkx
ƒ$»½*¾A¬
„ µ¶-· qjpQqc²
„
«Œ¬ Ï
ª
¤¿¦©¨
¬
­ sž Ÿp
r u w „
x
ƒ$»6½'¾A¬
„ µ¶-· qjpQqc²
„ÐÒÑ
u
Ï
¬
r u w w ¯
(A.45)
¨
Appendix A: Time dependent perturbation theory
136
Therefore, the total absorption cross section, Ó,ÖÔQÕc×Ô ØÙÚ"Û , equals (except for a constant
factor) the Fourier Transformation (FT) of the autocorrelation function Ü Ö ÙÝÛ . In
practice, the initial wavepacket (A.43) is propagated and the autocorrelation function is calculated according to equation (A.44). The spectrum is then obtained by
the Fourier Transformation of the autocorrelation function (cf equation (A.45)).
The initial condition (A.43) implies a spontaneous excitation ( Þ -pulse): The initial
(t=0) excited state wavefunction equals the wavefunction of the vibrational ground
state (ßáàmâ ) of the electronic ground state (ãäàmâ ) multiplied by the transition dipole function in the direction (å ) of the electric field, æäÖ8çYèSê é .
Spectrum in the case of bound-bound transitions
Here, a transition from the bound ground state to a bound excited state ë is considered. The initial wavepacket (A.43) at Ý à Ý ê àìâ is real and can be written as
a linear combination of the (real) eigenfunctions:
ï\ð
í Ö ÙÝ ê
àmâ Û àzîYï’ðòñ
ÙÝ ê àóâ Ûôõ
õ
ð×ö
Ù÷AøAÛ
(A.46)
The discrete eigenfunctions and values are defined by equation (A.4) and the completeness relation (A.7) holds. Using the wavepacket equation (A.11) for Ý à Ý ê à
â and Ý à Ý , the expression (A.44) of the autocorrelation function results in
Ü
ÙÝÛ àmùûú0üýÿþ î ï ð
ï ð
ñ
Ù Ý ê ó
à â Û¼Ùcô
ð×ö
õ
ٖ÷¼øbÛÛ þ î ð
õ
ñ õ
Ù Ý ê ó
à â Û å
×
ð
Ô ô
õ
î ð ñ
õ
õ
à
ð
ð
õ
ٖ÷AøAÛ ð
×
Ù â Û å
ð×)ö
õ
Ô
(A.47)
From the last expression it is obvious that the autocorrelation fulfills the symmetry
relation
Ü Ù ÝÛ àzÜ ÙÝÛ
(A.48)
which guarantees that the absorption spectrum (A.45) is real. This is a consequence of the fact that the initial wavepacket (A.43) is real. Propagating till the
autocorrelation function is essentially zero, assures that the calculated absorption
cross section is positive [41].
According to equation (A.45) the resulting absorption spectrum is
ÖQÔ Õc×Ô Ø ÙMÚ"Û àmñ Ý "!
Ó àmñ Ý "!
ÕcÔQÕ$#
ÕcÔQÕ$#
ú Ý î ð
6ù%'&
6î ð
õ
Ù â Û å
×
ð
Ô å
õ
&
ñ õ
ñ õ
ð
ð
Ù â Û ù%'&
&
× ú Ý å ç ð
é Ô
×
Ô
Appendix A: Time dependent perturbation theory
@BAGFIHGJ
(*)+-,/.10325476"89;:<9$=>2?;@BADCE)
137
C K 2MLOQ
N P
F
@BA5JU
4SRT4
(A.49)
where definition (A.40) of the P -function was applied.
Therefore, the absorption
@ A
4
and a height proportional
spectrum
consists of discrete lines with energy
@ A F;HVJ
@ A F J to
CE)
C K . In the case where WXZY is a constant (Condon approximation), the C )
0 CK
are equal to C WXZY C K times the so-called Franck-Condon factors. The result (A.49)
(except for a factor) can be also achieved by a population analysis of the initial
wavefunction (if the eigenfunctions and values are known).
Problem of light vs. molecule orientation
Normally the molecule is not oriented and it has to be classically averaged over
the different laser polarizations which implies a calculation of three spectra for x-,
[
[
[
[
y- and z-polarized light:
<: 9$:
(^_ ]
XZY\
Fa`bJc
<: 9$:
XZY\
<: 9$:
XdYe\
_]
FafgJhc
_]
<: 9;:
XZY\
F;iVJj
(A.50)
If, however, there is only one dominant component of the transition dipole moment in the Franck-Condon window (e.g. W/XdYlk mon W XZYpk qrs ), the transition dipole
moment function can be replaced by its modulus at the Franck-Condon point (because the modulus is approximately equal to the dominant component).
C WXZY C(ut
W KXdvk m
c
W KXdvk q
c
W KXdvk s
(A.51)
Using a constant factor instead of the transition dipole moment function is known
as the Condon approximation.
If more than one electronic excited (final) state is involved, then, according to
equation (A.38), the spectrum equals the sum of the individual spectra of the states
w
:
[
[
<: 9$:
Ye\
FyxzJ
)+,.1032476"8"9$:<9$=}2 ~
O
x7™
<: 9;:
XZY\
({?
X
'€

‰ X
O
‚
x
(*?
)+,.10|24 6"8"9$:<9$=}2-~'€

€
X
0 ?
€
c
FyxzJ
F
F J
:
0„ƒ X 0 2… Y‡†ˆ (
‚
F J„J
:
ƒ X 0
2e… Y†-ˆ ({)+,.10“2476"8"9$:<9$=}2”}•—– ?
Š ‹
1
ŽIŽy‘ :’
Œ
X
F
F J„J˜DU
ƒ X 0
(A.52)
(
4 Y\
where
. This means that in the case of several final electronic
states the Fourier
F J Transformation
F
F J„J (FT) of the sum of the individual autocorrelation
functions, ƒb:<9;: 0 (Sš X ƒ X 0 leads to the total electronic absorption spectrum.