Two-photon spectroscopy of the low lying Rydberg states of NO. I. The 3p and 3d complexes

Two-photon spectroscopy of the low lying Rydberg states of NO. I. The 3p and 3d

complexes

Henning Meyer

Citation: The Journal of Chemical Physics 107, 7721 (1997); doi: 10.1063/1.475087

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Two-photon spectroscopy of the low lying Rydberg states of NO. I.

The 3p and 3d complexes

Henning Meyer

Department of Physics and Astronomy, The University of Georgia, Athens, Georgia 30602-2451

͑Received 15 July 1997; accepted 14 August 1997͒

The rotational structure and polarization dependence of two-photon spectra of aligned ensembles of

open shell diatomics is investigated in terms of the spherical tensor components of the two-photon

absorption operator. The formalism allows the straightforward incorporation of state interactions

and perturbations. It is applied to the two-photon spectroscopy of NO, in particular to the excitation

of the Rydberg states derived from the 3p and 3d complexes. All states investigated show a nearly

quadratic power dependence indicating the saturation of the ionization step. Transitions dominated

2

by a zeroth rank tensor component ͑e.g., C ⌸–X ⌸ or H ⌺, H ²⌸–X ⌸͒ are insensitive to

Ј

a possible angular momentum alignment in the ensemble. These transitions are ideally suited to

determine degeneracy averaged observables, e.g., collision cross sections in a molecular beam

scattering experiment or product velocity anisotropies in a single color photodissociation

experiment. Rotational alignment data must be determined using two-photon transitions which are

2

Institute of Physics. ͓S0021-9606͑97͒03043-2͔

I. INTRODUCTION

have been studied extensively using primarily one-photon

excitation to the A ⌺ state in combination with ﬂuorescence

2

or ionization detection.^14–17When a single photon excitation

process is combined with ionization detection, the extraction

of population and alignment data often is severely compli-

cated due to partial saturation of the resonant excitation

step.^18–20On the other hand, many of the low lying Rydberg

states are easily accessible through efﬁcient two-photon ex-

citation processes which are much less troubled by saturation

phenomena. Successful application of these (2ϩ1) REMPI

schemes to product detection requires a thorough under-

standing of the saturation behavior and the two-photon spec-

troscopy including a possible laser polarization dependence

due to rotational angular momentum alignment or orienta-

tion. From the analysis of the different one-photon spectra,

many of the excited states have been shown to interact more

or less strongly with each other.²¹

During the past decade, substantial progress has been

made in the experimental study of molecular interactions.

Especially, the combination of molecular beams with the la-

ser spectroscopic preparation of initial reactant states and/or

state speciﬁc product detection has made possible the study

of different processes like photodissociation, energy transfer

and molecular reactions on a truly microscopic level. In ad-

dition to the achieved state resolution, laser spectroscopic

techniques are intrinsically sensitive to the orientational an-

isotropy of the angular momentum distribution, i.e., the

alignment or orientation.^1,2The resulting complications as

well as the possible beneﬁts have been discussed

extensively.³

Very recently, several laser based techniques have been

developed which aim at the simultaneous determination of

state and velocity resolved product distributions.⁴In one

class of experiments, velocity resolution is achieved by mea-

suring Doppler proﬁles in combination with ﬂuorescence or

direct absorption detection.^5–7In a second type of experi-

ments, resonance enhanced multiphoton ionization ͑REMPI͒

of the products is applied. The product velocity is deter-

mined through either spatial analysis of the ion displacement

In order to employ two-

photon excitation schemes which involve these excited

states, a general framework is needed which allows a

straightforward incorporation of these perturbations as well

as the determination of the rotational structure and the polar-

ization dependence.

Several authors have investigated the theory of nonreso-

nant two-photon excitation over the past two decades. The

case of an isotropic ensemble of open shell diatomic mol-

ecules has been treated by Bray and Hochstrasser²²as well as

Halpern et al.²³The application of spherical tensors to the

two-photon excitation problem has been pioneered by Mc-

Clain and Harris who treated the case of an isotropic en-

semble of symmetric top molecules.²⁴Similarly, Dixon et al.

employed spherical tensor components to describe the three-

photon excitation in isotropic molecular ensembles.²⁵The

probing of aligned ensembles of closed shell diatomics, sym-

metric top or asymmetric top molecules has been analyzed

recently for nonresonant two-photon^26–28or three-photon

8,9

͑ion imaging͒ or temporal analysis of the ion arrival time

͑ion time-of-ﬂight analysis͒.^10–13Various ionization schemes

have been employed in these experiments. Of particular im-

portance are so-called (2ϩ1) REMPI processes in which the

product molecule undergoes a two-photon excitation to an

intermediate electronic state followed by the absorption of a

third photon resulting in the ionization of the molecule. Us-

ing commercially available laser systems, even small mol-

ecules can be detected with high sensitivity.

Not too surprisingly, open shell atoms or diatoms are of

special interest in the ﬁeld of molecular reaction dynamics.

Beside the OH radical, systems involving the radical NO

J. Chem. Phys. 107 (19), 15 November 1997

0021-9606/97/107(19)/7721/11/$10.00

7721

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7722

Henning Meyer: Low lying Rydberg states of NO. I

excitation.²⁹Here, we describe the extension towards the

treatment of an aligned ensemble of open shell diatomic mol-

ecules. The formalism allows a straightforward incorporation

of possible electronic perturbations which might severely af-

fect the observed line strength.

As an application, we investigate the two-photon spec-

troscopy of several vibrational bands associated with the NO

Rydberg states derived from the 3p and 3d complexes. The

identiﬁed spherical tensor components in combination with

the saturation behavior allow the selection of a particular

REMPI detection scheme suitable to different types of ex-

periments. Polarization dependent intensities are expected

for those schemes which involve a two-photon transition car-

ried by a second rank tensor component, e.g., transitions to

2

the states D ⌺ or F ⌬. On the other hand, degeneracy

averaged quantities are accessed directly using REMPI

schemes with an intermediate Rydberg state whose two-

photon spectroscopy is dominated by a zeroth rank absorp-

FIG. 1. Orthogonal laser excitation geometry. The probed angular momen-

tum distribution is assumed to be cylindrically symmetric around the z axis.

The linearly polarized laser beam propagates along the y axis. The direction

of polarization is speciﬁed by the angle ␤.

2

tion tensor, e.g., transitions to the states C ⌸ or H ⌸.

These latter detection schemes are also suitable for single

color photodissociation experiments. Because the polariza-

tion vector of the photolysis and the probe laser light are

necessarily parallel in these experiments, different contribu-

tions due to the anisotropy in fragment velocity and due to

the rotational angular momentum alignment cannot be disen-

tangled. Meaningful velocity anisotropies can be extracted

from experimental data only if the detection scheme is not

sensitive to a possible alignment.

Another important advantage of the spherical tensor

treatment results from the simple transformation properties

of the latter under rotation. As discussed in detail in an ac-

companying paper, the tensor components characteristic to a

speciﬁc two-photon transition in the bare NO molecule can

be utilized ͑after the appropriate transformation͒ to explain

the rotational structure of the two-photon spectra associated

with the open shell NO–Ar complex.³⁰

2

˜

S_if

Tϭn J ,␶ ͒

␳

͉

.

͑1͒

͑

͚

i

M M

i

M

,M

f

i

The distribution over the different magnetic sublevels

denoted by M_iare expressed as the diagonal elements of a

density matrix ␳ whose trace is normalized to unity. Follow-

ing the deﬁnition of Ref. 31, these diagonal elements are

related to the state multipole moments T₀^(Q)

J_i

Q

0

J ϩM

i

ͱ_2Qϩ1

␳

ϭ

i

Ϫ͒

͑

ͩ

ͪ

͚

M M

i

M_iϪM_i

Q

͑Q͒

ϫT₀J ,␶ ͒.

͑2͒

͑

i

˜

The two-photon line strength S_ifis related to the matrix

elements of the components of a two-photon transition op-

erator T and a polarization tensor P. Expressing both quan-

(j)

tities as spherical tensors with components T_mand P⁽_m^j)de-

˜

ﬁned in a laboratory ﬁxed frame, the line strength takes the

form

II. THEORY OF NONRESONANT TWO-PHOTON

EXCITATION

͑ j͒ ͑ j͒

˜

In this section, we summarize brieﬂy the theory of non-

resonant two-photon absorption in an ensemble of open shell

diatomics characterized by cylindrical symmetry. We restrict

ourselves to an aligned ensemble, i.e., we assume that the

angular momentum distribution is invariant under reﬂection

with respect to a plane perpendicular to the axis of cylindri-

cal symmetry. Product ensembles of this type are encoun-

tered in molecular beam scattering and photodissociation ex-

periments. Typically, these ensembles are probed using

linearly polarized light in an orthogonal arrangement as

shown in Fig. 1. The intensity of a speciﬁc transition is pro-

portional to the sum of the square of the transition moments

between different magnetic sublevels M_iand M _fweighted

with the relative population of the initial M_ispeciﬁc state

and the overall population n(J_i,␶_i) of the rovibronic level

probed

S ϭ

f

͉

T_mP_m

͉

i .

͑3͒

͗

͘

͚

if

jm

The components of the polarization tensor are con-

structed from the polarization vector e which in the case of

linearly polarized light is given by

eϭ sin ␤,0,cos ␤͒.

͑4͒

͑

The angle ␤ speciﬁes the direction of the polarization

vector with respect to the axis of cylindrical symmetry. Us-

ing the Clebsch–Gordan series for the elements of the rota-

tion matrix, its components can be cast into the form

1

0

1

0

j

͑ j͒

_ϭͱ_4␲

P_m

Y

␤,0͒.

͑5͒

͑

ͩ

ͪ

jm

0

In order to determine the dependence of the transition

matrix elements on the rotational quantum numbers, we re-

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Henning Meyer: Low lying Rydberg states of NO. I

7723

1

2

alize that the overall wave functions contain an electronic

ͱ _{2J ϩ1͒ 2J ϩ1͒}

͑

S ⌳ ,⌳ ;j͒ϭ

͑

f

i

f

contribution which is best described in a molecule-ﬁxed co-

ordinate frame. Its orientation with respect to a laboratory

ﬁxed frame is speciﬁed by a set of Euler angles ␻. Thus the

two-photon transition operator deﬁned in the laboratory

frame is regarded as resulting from a similarity transforma-

T

Ϫ⍀ Ϫk

J ϩJ

i f

i

ϫ

Ϫ͒

1Ϫ⑀ ⑀ ⑀ Ϫ͒

͒

͑

͚

i

f

k

J_f

j

J_i

͑ j͒

ϫ

⌳

͉

T_k

͉

⌳

͗

͘

i

ͩ

ͪ

ͭ

f

⍀

Ϫk Ϫ⍀_i

tion of a body-ﬁxed transition operator involving the rotation

f

(j)

mk

operator R(␻) whose matrix elements are denoted D (␻)

͑

j͒

J ϩJ

i

f

ϩ⑀ Ϫ͒

Ϫ⌳

͉

T_k

͉⌳

͘

i

͑

͗

f

J_f

j

k

J_i

ϫ

͑10͒

ͩ

ͪ

͑ j͒ ͑ j͒*

͑ j͒

m

ͮ

˜

⍀

S ϭ

f

͉

T_kD_mk␻͒P

͉

i .

͘

͑6͒

͑

͗

͚ ͚

f

i

if

jm

k

It is obvious from Eq. ͑10͒ that for a speciﬁc transition

between to different rotational levels only one of the two ⌳

doublet components can be probed

In contrast to Eq. ͑3͒, the electronic parts of the wave

functions and and in Eq. ͑6͒ are now deﬁned with

respect to the molecule ﬁxed coordinate frame. For the open

shell diatomic NO molecule, the wave functions are ex-

pressed most conveniently in a Hund’s case ͑a͒ basis set

which diagonalizes the operators L_z, S_z, J², and J_z

͉

i

͉f

͘

T

k

J ϩJ

⌬J

i

1Ϫ⑀ ⑀ ⑀ Ϫ͒

^fϭ2⇔⑀ ϭ⑀ ⑀ Ϫ͒

͑

i

f

i

f

k

1

with J_fϭJ_iϩ⌬J.

͑11͒

͉

␩ J M ⍀ ⑀ ϭ

͉␩ ⌳ ⌺ ͉J ⍀ M

͕

͘ ͘

i i i i i i

͘

i

&

ϩ⑀_i

͉

␩ Ϫ⌳ Ϫ⌺

͉

J Ϫ⍀ M .

͘

͖

i

͑7͒

͘

i

Since the condition in Eq. ͑11͒ for a nonvanishing tran-

sition probability depends on ⌬J only, rotational lines of the

Q, O, or S branches can be used to probe the same ⌳ dou-

blet component ⑀_iof a speciﬁc initial rotational level, e.g., of

the electronic ground state. The other component must be

probed on R- or P-branch transitions.

To allow for the most general case, we must regard the

complete wave function involved in a particular electronic

system as the linear superposition of different basis states of

the type given in Eq. ͑7͒ which might involve different in-

teracting vibronic zeroth order states labeled by ␩

The symmetry quantum number ⑀ can take the values

Ϯ1 and is related to the overall parity by ⑀(Ϫ)^JϪSϩs͑i.e.,

Sϭ1/2 and sϭ0 for a ²⌸ system͒.³²The different basis

states are found by choosing a positive value for ⌳ and let-

ting the spin projection quantum number ⌺ take the values

Ϯ1/2. In this way also the quantum number ⍀ is restricted to

positive values. Since the two-photon transition operator

˜

does not act on the spin variables, the S_ifwill involve elec-

tronic matrix elements of the form Ϯ⌳ ͉ ͉Ϯ⌳ . The

T⁽_k^j)

͘

f i

͗

electronic wave functions as well as the body-ﬁxed represen-

tation of the T⁽_k^j)contain complex valued contributions only

through phase factors which depend on the azimuthal angle

of the individual electrons. For the matrix element not to

vanish, the different phases must cancel each other yielding

the relation

͉

iJ ␶ ϭ

Cⁱ

͉

␩

J M ⍀ ⑀ .

͑12͒

_iv_i

͘

͚

i

i i

i

The intensity of the two photon transition in an open

shell diatomic molecule is then given by the expression

⌳ ϭ⑀^TϪ⌳

͉

T_Ϫk

͉

Ϫ⌳ with

͉

⑀

͉

ϭ1. ͑8͒

͑ j͒

T

k

⌳

͉

T_k

͉

͗

͘

͗

͘

f

i

f

i

k

Applying Eq. ͑8͒, we derive the following expression for

the transition matrix element involving two Hund’s case ͑a͒

basis functions:

i

f

1

f

₂˜ ˜_*

.

f

2

1

2

Iϭn J ,␶ ͒

␳

C C C C S_iS_i

͑

͚

i

M M

f

1 1

i

2

M

,M

i i f f

1 2 1 2

i

f

͑13͒

J_f

j

J_i

͑ j͒

M ϩm

˜

i

S ϭ

P

Ϫ͒

S ⌳ ,⌳ ;j͒,

͑

f i

͑

ͩ

ͪ

Since J and M are necessarily good quantum numbers,

only basis states with the same values for M and J contribute

to the expansion in Eq. ͑12͒. This allows us to evaluate the

summations over the quantum numbers M_i, M _f, m₁and m₂

analytically.³³After some manipulations and rearrangements

of the occurring 3j-symbols, we arrive at the ﬁnal result

͚

if

m

M _fϪm ϪM_i

jm

͑9͒

where we have deﬁned the two-photon transition matrix el-

ement S(⌳_i,⌳_f;j)

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7724

Henning Meyer: Low lying Rydberg states of NO. I

TABLE I. Nonvanishing spherical tensor components T⁽_k^j)of the two-

photon absorption operator for transitions from the X ²⌸ state to various

Rydberg states.

J ϩJ

i

f

Iϭn J ,␶ ͒ Ϫ͒

2J ϩ1͒ 2J ϩ1͒

͑

i f

͑

i

j₁j₂

Q

0

͑Q͒

ͱ

ϫ

2Qϩ1T₀P_Qcos ␤͒

͑

ͩ

ͪ

͚

0

Q

j j

1 2

⍀

⌳

j

k

2

X

⌸

1/2, 3/2

1

0

1

0

j₁

0

1

0

1

0

j₂

0

ϫ

ͩ

ͪͩ

ͪ

2

D,H

C,H

⌺

⌸

1/2

1/2, 3/2

0

1

2

0

2

Ϫ1

0

0,Ϫ2

ϩ1

ϩ2

2

Ј

j₂J_iJ_f

ͱ

2j₁ϩ1 2j₂ϩ1

ͭ

ͮ

2

J_ij₁

Q

F

⌬

3/2, 5/2

5/2, 7/2

2

3

2

⌽

i

f

˜

2

˜

͑

i

1

2

1

*

ϫ

C C C C S ⌳ ,⌳ , j ͒S ⌳ ,⌳ ;j ͒ .

͑

͚

i

f

1

f

2

1

2

i i

1 2

f f

1 2

͑14͒

͑ j͒

⌳

͉

T_k

͉

⌳ ϭ ⌳ ϩk

͉

T_k

͉

⌳ ␦

,

⌳ ͑kϩ⌳ ͒

i

͗

͘

͗

͘

f

i

f

͑16͒

͑ j͒

Ϫ⌳

͉

T_k

͉

⌳ ϭ ⌳ ϩk

͉

T_k

͉

⌳ ␦

.

⌳ Ϫ͑kϩ⌳ ͒

i

͘

͗

͘

f

i

As expected, only tensor components with ranks jϭ0 or

f

2 contribute to the intensity of a spectral line when linearly

polarized light of a single frequency is used. An interference

term due to components belonging to tensors of different

rank is only possible for an anisotropic ensemble, i.e., in the

presence of nonvanishing state multipole moments with Q

Ͼ0. If the transition is carried exclusively by a zeroth rank

tensor component, then the intensity is solely determined by

the monopole moment of the angular momentum distribu-

tion. If such a transition is used to probe a molecular en-

semble, no polarization dependence is possible. Applied in a

scattering or photodissociation experiment, state resolved,

but degeneracy averaged dynamical quantities are accessible.

Another important class of experiments where the absence of

a polarization dependence will be advantageous is a single

color photodissociation experiment. Experimentally deter-

mined fragment velocity distributions will reﬂect solely the

velocity anisotropy although the ensemble might exhibit

strong rotational angular momentum alignment.

Because the orbital angular momentum quantum num-

bers ⌳_iand ⌳_fare chosen to be positive, we ﬁnd as a nec-

essary condition for a nonvanishing contribution of a tensor

component T_k^(j):kϭ⌳_fϪ⌳_ior kϭϪ(⌳_fϩ⌳_i). In the ab-

sence of perturbations, we expect two-photon transitions

2

from the NO ground state (X ⌸) to its low lying Rydberg

states to be carried by the tensor components listed in Table

I. Except for the ⌸ states where contributions due to tensors

of different rank are possible, all transitions to ⌺, ⌬, or ⌽

states will be carried by a single component of the second

rank tensor. Within these approximations the two-photon

spectra associated with the latter transitions will be uniquely

speciﬁed by a single electronic matrix element. Our experi-

mental data for the investigated transitions to several of the

low lying Rydberg states provide no evidence for other con-

tributions due to components of the two photon absorption

tensor which are not listed in Table I.

In an isotropic ensemble, the only nonvanishing multi-

pole moment is the monopole moment with Qϭ0 and Eq.

͑14͒ is simpliﬁed to

III. EXPERIMENT

Three different types of experiments have been per-

formed in order to probe ensembles of NO molecules at a

great variety of different temperatures. Extremely cold tem-

peratures in the order of about 2 K are found under molecu-

lar beam conditions. Details of the apparatus used have been

given elsewhere.³⁴Brieﬂy, molecular beam pulses are gener-

ated at a repetition rate of 10 Hz by expanding a gas mixture

of 5% NO in Ne into high vacuum. The pulses of about 40

␮s duration pass through a skimmer into a second differen-

tially pumped chamber where NO is ionized through (2

ϩ1) REMPI. Generated ions are mass selected in a time-of-

ﬂight mass spectrometer and detected with a multichannel

plate detector.

In a second type of experiment, we detect extremely hot

NO molecules resulting from the photodissociation of NO₂.

In this case, a mixture of 4% NO₂in Ne is used. The mixture

contains a very small amount of NO as an impurity which is

characterized by a rotational temperature of about 2 K. For

convenience, we perform single color experiments, i.e., the

same laser pulse is used as the photolysis laser as well as the

probe laser. Although the produced fragment ensemble can-

not uniquely be characterized by a rotational temperature,

2

n J ,␶ ͒

͑

1

0

1

0

j

0

i

Iϭ

ͩ

ͪ

͚

2J ϩ1͒

͑

j

i

f

2

˜

͑

i

1

2

1

*

ϫ

C C C C S ⌳ ,⌳ ;j͒S ⌳ ,⌳ ;j͒ .

͑

͚

i

f

2

1

2

i i

1 2

f f

1 2

͑15͒

In this case, the spectrum can be considered an algebraic

sum of independent spectra, each supported by its own ten-

sor component. The two-photon transition matrix elements

given in Eq. ͑10͒ not only determine the selection rules for

the total angular momentum and its projection, but they also

contain the electronic matrix elements Ϯ⌳ ͉ ͉⌳ . Ap-

T⁽_k^j)

͘

f i

͗

proximate selection rules for the orbital angular momentum

projection can be derived easily if we approximate the elec-

tronic wave function by a single electron wave function. In

this case, it can be shown that T⁽_k^j)ϳe^ik␾and we ﬁnd for the

two matrix elements in Eq. ͑10͒

J. Chem. Phys., Vol. 107, No. 19, 15 November 1997

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Henning Meyer: Low lying Rydberg states of NO. I

7725

these ‘‘hot’’ NO spectra are extremely useful in checking or

extending the spectroscopy of a particular Rydberg state to-

wards higher rotational and vibrational levels.

Spectra characterized by temperatures up to 300 K are

recorded in a third type of experiment.²⁸Here, the gas mix-

ture ͑5% NO in Ne͒ is expanded into a small vacuum cham-

ber. The chamber is pumped by a mechanical pump

equipped with a liquid nitrogen bafﬂe. A base pressure of

about 10^Ϫ2mbar is achieved with the pulsed molecular

beam valve operating. Again, NO molecules are ionized

through a (2ϩ1) REMPI process, and the ion current is

collected and ampliﬁed. Although these experiments are

technically less demanding, extreme care must be taken to

avoid saturation effects or ion avalanching.

The laser system consists of a grazing incidence dye

laser pumped by the second harmonic of a Nd:YAG laser.

The output of the dye laser is frequency doubled in a KDP

crystal. In the experiments reported here, we operated the

laser either on the dye DCM or Pyridine II which cover the

wavelengths range 620–680 or 690–740 nm, respectively.

For the experiments performed under high vacuum condi-

tions, the laser beam is focused with a 500 mm lens onto the

molecular beam while a 300 mm lens is used in those ex-

periments which rely on ion current detection. Typical pulse

energies of the second harmonic dye output range from 100

␮J to 1 mJ. The covered wavelength range enables us to

study two-photon transitions to various low lying Rydberg

states of NO. In particular, we studied the transitions

FIG. 2. Power dependence of the ion signal for excitation of the indicated

transitions. Panel A displays the result for direct excitation of NO using a

lens of 300 mm focal length. For panel B, NO is generated from the pho-

todissociation of NO₂͑500 mm focal length͒. Note, that both signals are

plotted as a function of laser pulse energy.

mixing for the ⌺ and ⌸ components in combination with

core penetration effects results in the irregular ordering of

the states derived from the 3d complex.⁴⁰

A. D ²⌺„v

؍

1…

Ј

Information about the rotational structure of the ﬁrst vi-

2

brational levels of the D ⌺ state has been derived from

direct vuv absorption data^41,42and near infrared emissions to

2

43

2

C ⌸( ϭ1)–X ⌸( ϭ0) and D ⌺( ϭ1)–X ⌸(

v

Љ

v

Ј

Љ

Ј

the A ⌺. More recently, also the near infrared emission

from the E ⌺⁴⁴and the H state⁴⁵to the ﬁrst two vibrational

2

ϭ0). Both excited states result from the formation of the 3p

2

complex. Similarly, the formation of the 3d complex gives

levels of the D ⌺ state has been observed. (2ϩ1) REMPI

2

rise to the three Rydberg states H ⌺, H ²⌸, and F ⌬.

2

Ј

detection via the D ⌺( ϭ0) level has been used by Mons

v

Ј

2

Here we report our investigation of their origin bands

through two-photon spectroscopy.

and Dimicoli for the detection of NO X ⌸( ϭ1) resulting

v

Љ

from the photodissociation of NO₂.¹¹

Although the excited vibronic levels D ⌺( ϭ1) and

2

v

Ј

2

the A ⌺( ϭ5) are known to interact, this perturbation

v

Ј

IV. TWO-PHOTON EXCITATION OF NO RYDBERG

STATES

39

causes only very small shifts in the order of 0.15 cm^Ϫ1

.

These shifts are clearly within the effective resolution of the

second harmonic output of our laser system at the two-

The one-photon absorption and emission spectra of the

NO molecule have been investigated extensively by many

groups. The different vibrational levels of the electronic

ground state have been studied using high resolution ir and

Fourier transform ir spectroscopy.^35,36The excited states of

the NO molecule can be classiﬁed into one of the two types:

valence type or Rydberg type.²¹The approximate series of

Rydberg states is derived from the formation of (n,1) com-

plexes characterized by the orbital angular momentum 1 of a

singly excited electron. Since the ground state of the NO^ϩ

ionic core is a ¹⌺ state, the character of the Rydberg states is

determined by the projection ␭ of the orbital angular momen-

tum, i.e., ␭ϭ⌳. The exact position of the derived states is

inﬂuenced critically by various interactions. The position of

the states derived from the p complexes (lϭ1) is determined

2

photon level. In practice, we ﬁnd the D ⌺(

v

Ј

2

ϭ1)–X ⌸( ϭ0) transition to be extremely well behaved.

v

Љ

As shown in Fig. 2͑a͒, the investigated rotational lines show

a nearly perfect quadratic power dependence; a prerequisite

for the applicability of the spectral simulation outlined in

Sec. II. The frequency spectra recorded under various mo-

lecular beam or jet expansion conditions are extremely well

described by the theory outlined in Sec. II. For the spectral

simulation in Fig. 3, only electronic wave functions associ-

2

ated with the D ⌺ state and a single nonvanishing compo-

(2)

Ϫ1

nent T

for the two-photon absorption tensor have been

included. The spectrum due to NO fragments resulting from

the single color photodissociation of NO₂is shown in Fig. 4.

Although we cannot expect to model the measured spectrum

correctly by assuming an ensemble characterized by a single

temperature, we ﬁnd fairly good agreement by assuming a

temperature of about 2000 K and weighting the population of

the excited spin–orbit component with an additional factor

0.6. Finally, energy conservation is enforced by including

by homogeneous interactions^37,38͑e.g., C ⌸–B ⌸͒ or het-

2

erogeneous interactions³⁹͑e.g., D ⌺–B ⌸͒ with the non-

2

Rydberg valence states ͑e.g., B ⌸ or B ²⌬͒. The large

2

Ј

quantum defect of the (nϩ1) s orbitals is responsible for a

near degeneracy with the nd orbitals. The resulting state

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7726

Henning Meyer: Low lying Rydberg states of NO. I

2

wavelength region of the D ⌺( ϭ1)–X ⌸( ϭ0) tran-

v

Љ

Ј

sition, we observe a nearly cubic power dependence of the

intensity for the various rotational lines probed ͓see Fig.

2͑b͔͒. For pulse energies lower than 500 ␮J ͑in combination

with a 500 mm focussing lens͒, we ﬁnd the unique situation

that the one photon absorption process in NO₂as well as the

two-photon resonant excitation step are negligibly saturated

in contrast to the ﬁnal ionization step.

B. C ²⌸„v

؍

1…

Ј

2

The rotational analysis of the ␦-system (C ⌸–X ⌸)

has been reported by Lagerquist and Miescher.³⁷These au-

thors also investigated the homogeneous perturbation due to

2

the non-Rydberg state B ⌸ treating both excited states as

2

pure Hund’s case ͑b͒. In the case of the band C ⌸(

v

Ј

2

ϭ1)–X ⌸( ϭ0), the major perturbation is caused by the

v

FIG. 3. Comparison of an experimental spectrum for the

D

²⌺(

vЈ

Љ

ϭ1)–X ²⌸( ϭ0) transition with a simulated spectrum assuming a tem-

ϭ10 level of the B state. For the higher members of the

v

Ј

vЉ

2

perature of 230 K.

C ⌸–X ⌸ and B ⌸–X ⌸ band systems these authors

included the spin–orbit interaction.⁴⁷Spin–orbit splittings

2

for the lower vibrational levels ( Ͻ7) of the B ⌸ state

v

Ј

have been determined by Engleman and Rouse.⁴⁸In their

experiments, the measured linewidths of about 1 cm^Ϫ1are

Doppler limited indicating the absence of severe predissocia-

tion. The predissociation of higher rotational levels of the

only those lines which probe rotational levels for which a

positive translational energy results. A bond energy of

25 130 cm^Ϫ1is used for the energy balance.⁴⁶While the

overall agreement is good, deviations in many details pro-

vide clear indications for some nonstatistical aspects of the

dissociation process. We also note, that the individual lines

do not necessarily represent the population of the probed

level. Because the transition is carried by a second rank ten-

sor component, a polarization dependence will result from a

possible rotational angular momentum alignment. For a

single laser photodissociation experiment, photolysis and

probe laser polarizations are necessarily parallel. Conse-

quently, the measured intensities ͑velocity unresolved͒ will

depend on the quadrupole moment A⁽₀²⁾of the angular mo-

mentum distribution. For the photodissociation of NO₂in the

2

perturbing pair C ⌸( ϭ0), B ⌸( ϭ7), has been stud-

v

Ј

ied experimentally by Rottke and Zacharias using ﬂuores-

cence excitation and two color resonance enhanced ioniza-

tion ͑(1ϩ1 ) REMPI͒.⁴⁹Single rotational state lifetimes in

Ј

2

the order of 1–10 ns (C ⌸) and 100–500 ns (B ⌸) are

reported.

Two-photon spectra have been measured for the

C ⌸( ϭ0)–X ⌸( ϭ0) transition using ﬂuorescence⁵⁰

2

v

Љ

Ј

or ionization detection.⁵¹In both cases, rotational contours

were simulated taking into account different possible two-

photon routes. While this method is based on different values

of the oscillator strengths for possible one-photon excita-

tions, in practice, the number of intermediate states included

has to be restricted, very often to a single one. Therefore,

measured two-photon spectra are reproduced only qualita-

tively with this method. A close inspection of individual ro-

tational branches reveals substantial deﬁciencies.

2

The electronic ground state X ⌸ of the NO molecule

must be treated as an intermediate between Hund’s cases ͑a͒

and ͑b͒. For the excited Rydberg states, a pure Hund’s case

͑b͒ angular momentum coupling scheme provides a very

good ﬁrst approximation. On the other hand, the theory out-

lined in Sec. II uses a Hund’s case ͑a͒ basis set for the de-

scription of the initial and the ﬁnal states connected by the

two-photon transition operator. In order to describe the ex-

cited state wave functions in terms of a Hund’s case ͑a͒ basis

set, we treat the unperturbed electronic states as belonging to

an intermediate case coupling scheme, i.e., we allow for

spin-uncoupling and spin–orbit interactions. In a second

step, the two states are assumed to interact with each other

FIG. 4. The (2ϩ1) REMPI spectrum of NO resulting from the photodisso-

ciation of NO₂in the region of the D ²⌺( ϭ1)–X ²⌸( ϭ0) transition.

The simulated spectrum was calculated assuming a rotational temperature of

with a constant interaction matrix element H .³⁷Using the

Ј

vЈ

vЉ

frequencies reported in Ref. 37, a set of molecular constants

is determined in a simultaneous least squares ﬁt of the lines

2000 K.

J. Chem. Phys., Vol. 107, No. 19, 15 November 1997

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Henning Meyer: Low lying Rydberg states of NO. I

TABLE II. Molecular constants ͑in cm^Ϫ1͒ used in the simulation of the

7727

spectra for transitions to different Rydberg states. Perturbing levels are

marked with an asterisk.

2

X

vЉ

⌸

X

vЉ

⌸

D

vЈ

⌺

(

ϭ0)³⁵

(

ϭ2)³⁶

(

ϭ1)

␯

A_s.o.

B_rot

3723.85

122.635

55 571.0

00

123.139 3

1.696 11

5.3

1.660 96

5.5

1.970

5.66

D

_rot/10^Ϫ6

2

C

⌸

B

⌸

C

⌸

B

⌸

*

(

ϭ1)

(

ϭ10)

(

ϭ2)

( ϭ12)

vЈ

␯

54 699.80

56 959.76

00

C

H

A_s.o.

B_rot

q_x

Ϫ406.530

80.633

58.055

3.90

1.958 6

0.012 46

157.783

Ϫ2.60

1.8141

Ј

46.05

0.9984

41.97

0.9918

FIG. 5. Power dependence ͑panel A͒ and frequency spectrum ͑panel B͒ for

the transition C ²⌸( ϭ1)–X ²⌸( ϭ0). The individual contributions

vЈ

vЉ

0.009 75

due to a zeroth rank and second rank component of the two-photon absorp-

tion tensor are shown also in panel B.

2

of a C ⌸( )–X ⌸( ϭ0) band and its perturbing

includes an additional contribution due to a T⁽₀²⁾component.

This term gives rise to additional branches of O, P, R, and

S type. Adjusting the amplitude of these two contributions,

we ﬁnd an optimum agreement for a ratio T²₀/T₀⁰ϭ0.8. No-

tice that the actual contribution in the spectrum due to the T₀²

component is only about 25%, because the intensity is deter-

mined by the square of the matrix element. An additional

scaling factor of 2/5 is introduced due to the square of the 3j

symbol in Eq. ͑15͒. Finally, in Fig. 6, we display the results

for the single color photodissociation in the wavelength re-

v

Љ

Ј

2

B ⌸( )–X ⌸ band. The standard deviation of the ﬁt is

v

Ј

about 1.0 cm^Ϫ1. For the C ⌸( ϭ1)–X ⌸( ϭ0) band,

2

v

Љ

Ј

larger deviations up to several wave numbers are found for

lines involving rotational levels around J ϭ19.5 where the

perturbation is strongest. The resulting molecular constants

are listed in Table II. It is interesting to note that the spin–

orbit splitting for the vibrational level ϭ2 of the C ⌸

state is inverted. Inverted splittings have also been reported

for higher vibrational levels. The spin–orbit constants found

for the two members of the B ⌸–X ⌸ systems are in good

agreement with extrapolated data based on the lower vibra-

tional levels B ( ϭ0Ϫ6).⁴⁷As demonstrated by Lager-

Ј

2

v

Ј

2

gion of the C ⌸( ϭ1)–X ⌸( ϭ0) band. Here, rota-

v

Љ

Ј

tional levels up to J ϭ30.5 are populated allowing the for-

Ј

v

Ј

mation of a sharp center band. This unique structure is a

quist and Miescher, a second deperturbation procedure re-

quires the inclusion of the interaction with all vibrational

levels of the B state.³⁷Although we could expect an im-

proved ﬁt of the transition frequencies using this extended

deperturbation scheme, the accuracy achieved in the ﬁrst or-

der deperturbation is only slightly worse than the effective

resolution achieved in our experiments as illustrated by the

simulated spectra in Figs. 5 and 6. Nevertheless, it is impor-

tant to realize that the rotationless origins of the various

2

members of the C ⌸–X ⌸ system depend critically on the

level of deperturbation. Especially the deperturbed position

2

of the origin of the C ⌸( ϭ2)–X ⌸( ϭ0) transition

v

Љ

Ј

is shifted by an additional 150 cm^Ϫ1to the blue in compari-

son with the origin found from the ﬁrst order deperturbation.

Before we can apply the results of Sec. II to simulate the

rotational structure of an observed band, we must verify that

the ionization step is completely saturated while the resonant

two-photon excitation is not saturated. The nearly perfect

quadratic power dependence of the ion signal for different

rotational lines demonstrates that this is indeed the case. As

an example, we have displayed the measured power depen-

dence for the Q_2lc(4.5) line in Fig. 5͑a͒. In Fig. 5͑b͒ an

experimental spectrum measured at 175 K is compared with

different simulated spectra. The spectrum at the top, results

from a simulation assuming only a nonvanishing matrix ele-

ment due to a T⁽₀⁰⁾component while the bottom spectrum

FIG. 6. REMPI spectrum of NO resulting from the photodissociation of

NO₂in the region of the C ²⌸( ϭ1)–X ²⌸( ϭ0) transition ͑trace b͒.

vЈ

vЉ

Traces a, c, and d represent different simulations of the rotational structure.

The simulation shown as trace a does not include any inﬂuence due to the

homogeneous interaction with the B ²⌸( ϭ10) level. Trace d demon-

vЈ

strates the inﬂuence of the perturbation on the various line positions while

trace c also takes into account the effect of the perturbation on the line

strengths. Note that the assumed temperature for trace c and d differs by a

factor of 4.

J. Chem. Phys., Vol. 107, No. 19, 15 November 1997

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7728

Henning Meyer: Low lying Rydberg states of NO. I

FIG. 7. Effective rotational constant as a function of J for the various

vibrational levels of the C ²⌸ ͑thick lines͒ and B ²⌸ ͑thin lines͒ states. The

ﬁgure has been adapted from Ref. 37.

FIG. 8. Power dependence ͑panel A͒ and (2ϩ1) REMPI spectrum ͑panel

B͒ for the F ²⌬–X ²⌸ transition. The spectrum in panel B was recorded

under molecular beam conditions.

clear manifestation of the perturbation due to the ϭ10

v

Ј

c. In both cases, we adjusted the temperature to reproduce

qualitatively the measured spectrum. If we neglect the effect

of the perturbation on the intensity, a temperature of about

500 K is required to reproduce qualitatively the measured

intensity distribution. This result is in clear contrast to the

level of the B state. The simulated spectrum ͑trace a͒ shown

in the top part of Fig. 6 has been calculated assuming a

temperature of 500 K and the absence of any perturbation. In

this case, we ﬁnd the band head formation in the main

branches to the red reﬂecting the increased rotational con-

stant associated with a Rydberg state of NO. On the other

hand, if the perturbation is included, we see the formation of

additional band heads towards higher frequencies indicative

of a decreased rotational constant. The strong variation of the

effective rotational constants as a function of the energy

level is illustrated in Fig. 7 which we adapted from Ref. 37.

2

ﬁndings for the near D ⌺( ϭ1)–X ⌸( ϭ0) band

v

Љ

Ј

where a temperature of roughly 2000 K was found. On the

other hand, if the perturbation is included, a signiﬁcantly

higher temperature consistent with the results found for the

2

D ⌺( ϭ1)–X ⌸( ϭ0) band is found. For the one-

v

Љ

Ј

photon absorption, it is found that the various transitions to

2

the vibrational levels of the state B ⌸ derive their oscillator

In the energy region for rotational levels with 15.5ϽJ

2

Ј

strength exclusively from members of the C ⌸–X ⌸

2

system.⁵³In the extreme case of the perturbing pair

Ͻ20.5 the effective rotational constant within the C ⌸(

v

2

ϭ1)–X ⌸( ϭ0) band changes from one characteristic to

2

v

Љ

C ⌸( ϭ2)–B ⌸( ϭ12) the intensity is shared equally

v

Ј

a NO Rydberg state to a much smaller one typical for the B

between the two states. This is also observed for the corre-

sponding two-photon spectra ͑see Fig. 4 of Ref. 30͒. For the

entire wavelength range covered in this study, only few

state with its non-Rydberg character. Similarly, for the

2

C ⌸( ϭ2)–X ⌸( ϭ0) band, there is no range of J

v

Љ

Ј

values for which a constant effective rotational constant is

2

bands associated with transitions to the state B ⌸ have an

found. At small J values, the excited state vibrational level is

intensity comparable to the excitation of the Rydberg states.

The observed bands are heavily perturbed and thus gain their

2

perturbed strongly by the B ⌸( ϭ12) level while for

v

Ј

2

large J values the perturbation due to the B ⌸( ϭ13)

2

v

Ј

line strengths from their interaction with the C ⌸ state.

levels becomes important. The change in the rotational con-

stant is also accompanied by a signiﬁcant reduction in the

2

C. F

⌬

2

line strength. For the simulation of the C ⌸(

v

Ј

2

The one-photon vuv absorption spectrum of the

ϭ1)–X ⌸( ϭ0) band ͑trace c in Fig. 6͒, we assumed

v

Љ

54

2

F ⌬–X ⌸ transition was analyzed ﬁrst by Jungen. The

two-photon spectra of the 0–0 band recorded under different

expansion conditions are displayed in Figs. 8 and 9. Also for

this system, we ﬁnd a nearly perfect quadratic power depen-

that the electronic two-photon transition matrix element con-

necting the ground state and the unperturbed B state can be

neglected in comparison with the matrix element for the un-

perturbed C state.⁵²Consequently, the intensity of the lines

2

dence. The F ⌬ state is perturbed by the non-Rydberg state

within the C ⌸( ϭ1)–X ⌸( ϭ0) band are weighted

v

Љ

Ј

B

²⌬.⁵⁵The degree of perturbation strongly depends on the

with the square of the expansion coefﬁcient associated with

Ј

2

vibrational level. The lowest level F ⌬( ϭ0) is found to

v

the unperturbed C state wave function. Near J ϭ19.5 this

Ј

be perturbed by the level B ²⌬( ϭ2). Although this per-

v

Ј

coefﬁcient is signiﬁcantly reduced due to the perturbation

causing the decrease in the line strength. The implications

for the spectroscopic determination of populations can be

seen by comparing the two simulated spectra in the bottom

part of Fig. 6. Trace d has been calculated assuming no

intensity change due to the perturbation while the effect of

the perturbation on the intensity has been included for trace

Ј

turbation is very small, it has been included for the simula-

tion shown in the bottom part of Fig. 9. Within the frame-

work of Sec. II, all spectral features are accounted for by

assuming a single nonvanishing tensor component T and

using the molecular constants given in Table III. The con-

stants are derived in a simultaneous least squares ﬁt of the

(2)

ϩ1

J. Chem. Phys., Vol. 107, No. 19, 15 November 1997

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Henning Meyer: Low lying Rydberg states of NO. I

7729

FIG. 10. Power dependence ͑panel A͒ and (2ϩ1) REMPI spectrum ͑panel

FIG. 9. Comparison of the (2ϩ1) REMPI spectrum with a simulated spec-

trum at Tϭ210 K. At lower frequencies, the spectrum is overlapped by the

B

B͒ of the H ²⌺, H ²⌸–X ²⌸ transition.

Ј

²⌸( ϭ18)–X ²⌸( ϭ0) band.

vЈ vЉ

plex. In particular, we must take into account the interaction

2

between its 3d␴ and 3d␲ components. Although there is a

frequencies for the vuv transitions F ⌬( ϭ0)–X ⌸ and

v

Ј

2

signiﬁcant mixing between the ⌺ component and the E ⌺

B

²⌬( ϭ2)–X ⌸ ͑standard deviation of about

v

Ј

0.2 cm^Ϫ1͒.⁵⁶Apart from the extremely small spin-orbit split-

state derived from the 4s complex, we do not expect this

perturbation to seriously affect the rotational structure of the

2

ting found for the F ⌬ state, the constants are in good

two-photon spectrum of the H,H –X transition. The elec-

Ј

agreement with the ones reported originally by Jungen who

treated the state as a pure Hund’s case ͑b͒.⁵⁴

tronic matrix elements involving the two mixed components

will be very similar because both electronic state are essen-

tially pure Rydberg states. Since we are interested only in

transitions probing the two components of the H state, we

neglect the homogeneous perturbation and assume the pres-

2

D. H ²⌺,H

⌸

Ј

Of particular interest is the two-photon spectroscopy of

the H–X transition. As can be seen in Fig. 10͑a͒, a quadratic

power dependence of the ionization signal is found for pulse

energies smaller than 350 ␮J using a 300 mm focusing lens.

For larger pulse energies clear deviations are observed indi-

cating the onset of saturation. A comparison with the results

for the other Rydberg states indicates that the transition to

the H state is more easily saturated. The theoretical descrip-

tion presented in Sec. II can be applied to simulate those

two-photon spectra which have been recorded at low pulse

energies. Spectra recorded under different expansion condi-

tions are displayed in Figs. 10͑b͒ and 11 together with the

results of the simulation.

2

ence of a single effective H ⌺ component. The rotational

structure is dominated by the heterogeneous perturbation be-

tween the ⌸ and the ⌺ components due to the l-uncoupling

interaction. This interaction has been identiﬁed in the vuv

absorption spectrum by Huber and Miescher treating the ex-

In order to determine the excited state energy levels and

wave functions, we must take into account the different per-

turbations relevant for the states derived from the 3d com-

TABLE III. Molecular constants ͑in cm^Ϫ1͒ used in the simulation of the

spectra for transitions to different Rydberg states of the 3d complex. Per-

turbing levels are marked with an asterisk.

²⌬( ϭ0)

vЈ

B

²⌬( ϭ2)

vЈ

H

²⌸( ϭ0) ²⌺( ϭ0)

H

vЈ vЈ

*

F

Ј

␯

62 045.25

62 717.3

00

C

H

A_s.o.

B_rot

D

b

a_ϩ

Ϫ312.18

Ϫ13.842

Ј

26.17

0.02

1.970

Ϫ6.36

1.273

1.356

2.006

5.62

3.833

1.845

FIG. 11. Comparison of the (2ϩ1) REMPI spectrum with a simulated

spectrum at Tϭ175 K. The theoretical spectrum represents the superposi-

tion of two spectra due to a zeroth rank and a second rank tensor component.

The small contribution due to the second rank component is shown in the

inset ͑trace a: experimental, trace c: zeroth rank component only, trace b:

zeroth rank plus second rank component͒.

1.993

5.62

_rot/10^Ϫ6

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7730

Henning Meyer: Low lying Rydberg states of NO. I

cited state as a pure Hund’s case ͑b͒ type electronic state.⁵⁷

Molecular constants based on a Hund’s case ͑a͒ basis are

determined in a procedure similar to the one described for

of the experimental spectrum seems to be qualitatively well

reproduced, a closer inspection reveals quite substantial dif-

ferences in the intensities. Furthermore, the model clearly

produces several rotational branches which are absent from

the experimental spectrum. Most likely these branches are

due to an overestimation of the direct excitation to the ⌺

component. As stated above, we ﬁnd that this contribution is

negligible in comparison to the excitation of the ⌸ compo-

nent.

The dominance of the zeroth rank tensor component

makes REMPI detection of NO through the H state ideally

suited for those experiments which are designed to determine

degeneracy averaged observables, e.g., degeneracy averaged

but state resolved integral or differential scattering cross sec-

tions. Another important application is the NO product de-

tection in single color photodissociation experiments. In this

case information about the anisotropy of the velocity distri-

bution of the fragments can be derived without the interfer-

ence from a possible rotational angular momentum align-

ment.

2

the C ⌸ state. The heterogeneous perturbation between a ⌺

and a ⌸ state is treated in Ref. 58. Additional terms due to

centrifugal distortion have been included.⁵⁹The off-diagonal

elements of the resulting interaction matrix contain contribu-

tions due to spin–orbit, spin-electronic, and l-uncoupling in-

teractions expressed in terms of the interaction constants a_ϩ

and b. Since states of different parity do not interact, a 3

ϫ3 Hamiltonian matrix results for each parity. A complete

set of molecular parameters for the H,H state is derived

Ј

from a least squares ﬁt of the frequencies reported by Huber

and Miescher and listed in Table III. The ﬁt provides a stan-

dard deviation of about 0.5 cm^Ϫ1which is comparable to the

effective resolution in our experiment. As pointed out by

Bernard et al. the ﬁt could be improved by taking into ac-

2

count the heterogeneous interaction with the states F ⌬ and

B

²⌬.⁴⁵However, since we are primarily interested in the

Ј

reproduction of our two-photon spectra, it is sufﬁcient to use

the smaller basis set. In this case, the resulting eigenfunc-

tions are a superposition of the Hund’s case ͑a͒ basis func-

V. CONCLUSIONS

tions associated with the three components of the H,H state

Ј

A general framework is presented for the theoretical de-

scription of nonresonant two-photon transitions in open shell

diatomics in terms of components of spherical tensor opera-

tors. The model allows the straightforward incorporation of

perturbations between different electronic states. Further-

more, it allows for a possible polarization dependence of

individual rotational lines in the case of an aligned ensemble

of molecules. The rotational structure of several two-photon

spectra involving the low lying Rydberg states of NO have

been simulated and found to be in excellent agreement with

the measured spectra. The nearly perfect quadratic power

dependence found for all states studied provides clear experi-

mental evidence for the saturation of the ﬁnal ionization step.

The relevant tensor components contributing to the two-

photon line strength have been identiﬁed for these transi-

1/2

͉

H␶JM⑀_HϭC^⌺

͉

H,01/2JM⑀_HϩC^⌸

͉

H ,11/2JM⑀

͘

H

Ј

͘

ϩC^⌸

H ,13/2JM⑀

Ј

.

͘

H

3/2

͉

According to Sec. II, two-photon transitions from the

electronic ground state to the different basis states will be

(2)

Ϫ1

carried by a T component for the ⌺ state and a T⁽₀⁰⁾or T₀⁽²⁾

component for the ⌸ state. In practice, we ﬁnd that the mea-

sured two-photon spectrum can be accounted for completely

by the assumption of a dominant zeroth rank tensor compo-

nent and a minor second rank tensor component T⁽₀²⁾. The

electronic matrix element associated with the latter is

roughly a factor of 2.5 smaller than the one resulting from

the T⁽₀⁰⁾component. The effective contribution to the spec-

trum of the minor component is only 6% as can be seen in

the inset in Fig. 11. Trace c was calculated assuming only a

zeroth rank component which allows only lines belonging to

Q branches. Small additional lines appear in trace b due to

the second rank tensor component in agreement with the

experimental spectrum shown as trace a. No evidence for

contributions due to a direct excitation of the H ⌺ compo-

nent is found. The assumption of a nonvanishing T

tribution results in additional branches which are not ob-

served experimentally.

The H–X transition has been studied previously using

two-photon resonance excitation in combination with ﬂuo-

rescence or ionization detection. Zhao et al. used a Dunham-

type expansion to assign rovibrational levels.⁶⁰Vient et al.

combined the known molecular constants and one-photon

transition probabilities with a propensity rule model to simu-

2

tions. The H ⌺, H ²⌸–X ⌸ transition is found to be

Ј

carried exclusively by a zeroth rank tensor component. The

resulting absence of a polarization dependence in combina-

tion with its extraordinary strength makes this transition ide-

ally suitable for those investigations in which degeneracy

averaged quantities are of interest. Possible applications are

the determination of degeneracy averaged scattering cross

sections or the investigation of velocity anisotropies in single

color photodissociation experiments. On the other hand, in-

formation on the rotational angular momentum alignment is

accessible through those transitions which are carried by a

2

(2)

Ϫ1

con-

2

second rank tensor component ͑e.g., D ⌺–X ⌸,

2

F ⌬–X ⌸͒.

ACKNOWLEDGMENTS

late the rotational structure of the H,H –X two-photon

Ј

transition.⁶¹This model takes into account the different two-

The author would like to thank J. Miller for reading and

giving valuable remarks on the manuscript. Professor K.

Dressler did kindly provide spectroscopic data from the ar-

chives of the late Professor E. Miescher. Partial ﬁnancial

2

photon pathways via the intermediate states C ⌸ and

D ⌺. It is important to realize that no adjustable parameters

are required within this model. While the general appearance

2

J. Chem. Phys., Vol. 107, No. 19, 15 November 1997

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Henning Meyer: Low lying Rydberg states of NO. I

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J. Chem. Phys., Vol. 107, No. 19, 15 November 1997

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Article Doi

Two-photon spectroscopy of the low lying Rydberg states of NO. I. The 3p and 3d complexes

DOI: 10.1063/1.475087

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