E Gaidamauskas
email: erikas.gaidamauskas@tfai.vu.lt

C Nazé

P Rynkun

G Gaigalas

P Jönsson

M Godefroid

Tensorial form and matrix elements of the relativistic nuclear recoil operator

Keywords: numbers: 31.15.ac, 31.15.aj, 31.15.am, 31.15.V-, 31.30 32.10.Fn Isotope shifts, mass shift, relativistic nuclear recoil Friday

Within the lowest-order relativistic approximation (∼ v 2 /c 2 ) and to first order in m e /M , the tensorial form of the relativistic corrections of the nuclear recoil Hamiltonian is derived, opening interesting perspectives for calculating isotope shifts in the multiconfiguration Dirac-Hartree-Fock framework. Their calculation is illustrated for selected Li-, B-and C-like ions. The present work underlines the fact that the relativistic corrections to the nuclear recoil are definitively necessary for getting reliable isotope shift values.

Introduction

Nuclear and relativistic effects in atomic spectra are treated in the pioneer works of Stone [1,2] and Veseth [3]. The theory of the mass shift has then been reformulated by Palmer [4]. Calculations of nuclear motional effects in many-electron atoms have been performed by Parpia and co-workers [5,6] in the relativistic scheme, using fully relativistic wave functions, but adopting the non-relativistic form of the recoil operator. Relativistic nuclear recoil corrections to the energy levels of multicharged ions have been estimated by Shabaev and Artemyev [7] who derived the relativistic corrections of the recoil Hamiltonian. In a study of isotope shifts of forbidden transitions in Be-and B-like argon ions, Tupitsyn et al [8] showed that a proper evaluation of the mass isotope shift requires the use of this relativistic recoil operator. The latter has also been shown to be crucial by Porsev et al [9] for calculating isotope shifts of transitions between the fine structure energy levels of the ground multiplets of Fe I and Fe II.

As far as computational atomic structure is concerned, the extension of the available relativistic codes such as grasp2k [10] or mcdf-gme [START_REF] Indelicato | Mcdfgme, a multiconfiguration dirac-fock and general matrix elements program[END_REF][START_REF] Desclaux | Methods and Techniques in Computational Chemistry[END_REF] is needed for estimating these mass corrections properly for any many-electron system. Programs to calculate pure angular momentum coefficients for any scalar one-and two-particle operator are available [START_REF] Gaigalas | [END_REF] but do require the knowledge of the tensorial structure of the operators to be integrated between the many-electron atomic wave functions [START_REF] Grant | Relativistic Quantum Theory of Atoms and Molecules. Theory and Computation. Atomic, Optical and Plasma Physics[END_REF]. The tensorial form of the nuclear recoil Hamiltonian is derived in the present work, opening interesting perspectives for calculating isotope shifts in the multiconfiguration Dirac-Hartree-Fock (MCDHF) framework.

The relativistic mass shift operator

In the MCDHF method, the atomic state function (ASF) Ψ(γP JM J ), of a stationary state of an atom, is expressed as a linear combination of symmetry-adapted configuration state functions (CSFs) Φ(γ p P JM J ), i.e.

Ψ(γP JM

J ) = p c p Φ(γ p P JM J ), ( 1
)
where J is the total electronic angular momentum of the state, γ represents the electronic configuration and intermediate quantum numbers, and P stands for the parity. The mixing coefficients c p and the one-electron radial wave functions spanning the CSFs are optimized by solving the MCDHF equations iteratively until self-consistency. The latter are derived by applying the variational principle to the energy functional based on the Dirac-Coulomb Hamiltonian [START_REF] Grant | Relativistic Quantum Theory of Atoms and Molecules. Theory and Computation. Atomic, Optical and Plasma Physics[END_REF]

H DC = N i=1 cα i • p i + (β i -1)c 2 + V (r i ) + N i<j 1 r ij , (2)
where V (r i ) is the monopole part of the electron-nucleus interaction, α and β are the (4 × 4) Dirac matrices and c is the speed of light (c = 1/α in atomic units, where α is the fine-structure constant).

The mass shift of the energy levels in an atom with nuclear mass M is caused by the recoil motion of the atomic nucleus. The corresponding recoil Hamiltonian

H MS = 1 2M N i,j p i • p j - αZ r i α i + (α i • r i ) r i r 2 i • p j , (3)
has been derived within the lowest-order relativistic approximation and to first order in m/M by Shabaev and collaborators [7,8]. Rewriting it as the sum of the normal mass shift (NMS) and specific mass shift (SMS) contributions and using the tensorial form

r 1 = rC 1 , (3) becomes
H MS = H NMS + H SMS , (4)
with

H NMS = 1 2M N i=1 p 2 i - αZ r i α i • p i - αZ r i α i • C 1 i C 1 i • p i , (5)
H SMS = 1 M N i<j p i • p j - αZ r i α i • p j - αZ r i α i • C 1 i C 1 i • p j , (6)
that, in both cases, are rewritten as a sum of three separate contributions:

H NMS ≡ H 1 NMS + H 2 NMS + H 3 NMS , (7)
and

H SMS ≡ H 1 SMS + H 2 SMS + H 3 SMS . (8)
Since the expectation values of the NMS and SMS operators are evaluated with the MCDHF wave functions, the expectation values H 1 NMS and H 1 SMS partly contain the relativistic contributions. Tupitsyn et al [8] pointed out that averaging the nonrelativistic recoil operator with the relativistic wave functions strongly overestimates the relativistic correction to the recoil effect such as it becomes important to use the complete form (3) when one works in the relativistic scheme.

Normal mass shift expectation value

The (mass-independent) normal mass shift parameter K NMS is defined by the following expression

K NMS M ≡ Ψ(γP JM J )|H NMS |Ψ(γP JM J ) . (9)
By analogy with (7), we define K NMS as the sum of

K NMS = K 1 NMS + K 2 NMS + K 3 NMS .
Applying the Wigner-Eckart theorem [START_REF] Cowan | The Theory of Atomic Structure and Spectra[END_REF], the matrix element of the normal mass shift operator is M J -invariant and is proportional to the reduced matrix element (r.m.e.) ‡ Using multiconfiguration expansions (1), the reduced matrix elements of the general spherical tensor operator

K NMS M = 1 √ 2J + 1 Ψ(γP J) H NMS Ψ(γP J) = [Ψ(γP J) H NMS Ψ(γP J)] . (
T k q becomes Ψ(γP J) T k Ψ(γP J) = p,s c p c s Φ(γ p P J) T k Φ(γ s P J) . (11)
The reduced matrix elements of the one-electron operator T k = i t k (i) between CSFs is expressed as a sum over single-particle reduced matrix elements

Φ(γ p P J) T k Φ(γ s P J) = a,b T ps (ab) n a κ a t k n b κ b , (12)
where the T ps (ab) are the spin-angular coefficients arising from Racah's algebra [START_REF] Gaigalas | [END_REF][START_REF] Grant | Relativistic Quantum Theory of Atoms and Molecules. Theory and Computation. Atomic, Optical and Plasma Physics[END_REF][START_REF] Gaigalas | [END_REF]. Introducing the one-body normal mass shift operator associated to (5) (

H NMS = i h NMS (i)) h NMS = 1 2M p 2 - αZ r α + α • C 1 C 1 • p , (13)
we hereafter derive the expression of its r.m.e., using relativistic central-field one-electron wave functions

ψ na,κa,ma (r, σ) = 1 r P na,κa (r)Ω κa,ma (θ, φ, σ) iQ na,κa (r)Ω -κa,ma (θ, φ, σ) . (14)
P a and Q a are respectively the large and small components of the relativistic one-electron radial wave function a = (n a κ a ), where κ = (l -j)(2j + 1).

Introducing the notation ∂ r ≡ ∂ ∂r , the action of the operator p 2 on the large (F = P ) and the small (F = Q) component of a relativistic wave function

p 2 F n,κ (r) r Ω κ,m (θ, φ, σ) = 1 r -∂ 2 r + l(l + 1) r 2 F n,κ (r)Ω κ,m (θ, φ, σ) (15)
is found using

p 2 = -∆ = - 1 r 2 ∂ r r 2 ∂ r + l 2 r 2 . ( 16
)
From this expression and integrating by parts, the first term of the one-electron matrix element NMS operator (13) becomes,

n a κ a m a | p 2 2 |n b κ b m b = δ(κ a m a , κ b m b ) (17) × 1 2 ∞ 0 dr (∂ r P a ) (∂ r P b ) + (∂ r Q a ) (∂ r Q b ) + l b (l b + 1)P a P b + l b ( l b + 1)Q a Q b r 2 ,
with l = 2j -l. Building the Dirac matrices α from α = σ x ⊗ σ with

σ x = 0 1 1 0 , (18)
one rewrites the second and the third parts of the NMS operator [START_REF] Gaigalas | [END_REF] as

-αZ 2r α • p + α • C 1 C 1 • p ≡ σ x ⊗ A = 0 A A 0 (19) with A = -αZ 2r σ • p + σ • C 1 C 1 • p . (20)
Taking into account that (see (A.4.9) and (3.2.14) in [START_REF] Grant | Relativistic Quantum Theory of Atoms and Molecules. Theory and Computation. Atomic, Optical and Plasma Physics[END_REF])

σ • C 1 = σ • e r = σ r ; C 1 • p = e r • p = (-i∂ r ), (21)
and

σ • p = -iσ r ∂ r + K + 1 r , (22)
with

K = -(1 + σ • l), the operator A becomes A = -αZ 2r (-iσ r ) 2∂ r + K + 1 r . ( 23
)
Acting on the one-electron relativistic wave function component, it gives

A F n,κ (r) r Ω κ,m = -αZ 2r i r 2∂ r + κ -1 r F n,κ (r)Ω -κ,m , (24)
from which one derives, integrating by parts, the one-electron matrix element of the second and third parts of the NMS operator

n a κ a m a |σ x ⊗ A|n b κ b m b = δ(κ a m a , κ b m b ) (25) × 1 2 ∞ 0 (-2αZ) Q a ∂ r P b + Q b ∂ r P a r + (-αZ) κ b -1 r 2 (P b Q a + P a Q b ) dr
Combining [START_REF] Wigner | Group Theory and its Applications to the Quantum Mechanics of Atomic Spectra[END_REF] and (25) to deduce the r.m.e. of the normal mass shift operator [START_REF] Gaigalas | [END_REF], we obtain the final expression

[n a κ a h NMS n b κ b ] = δ(κ a , κ b ) (26) × 1 2M ∞ 0 (∂ r P a ) (∂ r P b ) + (∂ r Q a ) (∂ r Q b ) + l b (l b + 1)P a P b + l b ( l b + 1)Q a Q b r 2 + (-2αZ) Q a ∂ r P b + Q b ∂ r P a r + (-αZ) κ b -1 r 2 (Q a P b + Q b P a ) dr.

Specific mass shift expectation value

Similarly to (10), the (mass-independent) specific mass shift parameter K SMS is defined as

K SMS M ≡ [Ψ(γP J) H SMS Ψ(γP J)] , (27)
and K 1 SMS , K 2 SMS , K 3 SMS as its contributions according to (8). Its evaluation requires the calculation of the corresponding matrix elements in the CSF space. For the general scalar two-particle operator

G = i<j g(i, j) (28)
with

g(i, j) = k g k (r i , r j ) T k (i) • T k (j) , (29)
the reduction of the many-electron r.m.e. in terms of the two-electron integrals X k , also called effective interaction strengths [START_REF] Grant | Relativistic Quantum Theory of Atoms and Molecules. Theory and Computation. Atomic, Optical and Plasma Physics[END_REF],

[Φ(γ p P J)

i<j g(i, j) Φ(γ s P J)] = abcd k v (k) ps (abcd)X k (abcd) (30)
can be performed using Racah's algebra [START_REF] Gaigalas | [END_REF][START_REF] Grant | Relativistic Quantum Theory of Atoms and Molecules. Theory and Computation. Atomic, Optical and Plasma Physics[END_REF][START_REF] Gaigalas | [END_REF]. For the specific mass shift Hamiltonian (6), using k = 1, all three terms have the particular form

g(i, j) = g 1 (r i )g 1 (r j ) T 1 (i) • T 1 (j) (31)
in which the radial part g 1 (r i , r j ) of ( 29) is factorized. Adopting the covariant notation for the 3j-symbol of Wigner [START_REF] Wigner | Group Theory and its Applications to the Quantum Mechanics of Atomic Spectra[END_REF] and using the definition of the scalar product of two irreducible tensor operators and the Wigner-Eckart theorem, the matrix element of (31) can be written as follows

ab|g(i, j)|cd = 1 q=-1 m a 1 j c j a q m c m b q j d j b 1 m d X 1 (abcd), ( 32
)
where

X 1 (abcd) = -a g(r i )T 1 (i) c b g(r j )T 1 (j) d . (33)
From the structure of ( 8), the latter has three components

X 1 (abcd) = X 1 1 (abcd) + X 1 2 (abcd) + X 1 3 (abcd) (34)
that we analyzed hereafter separately.

2.2.1. First part: X 1 1 (abcd)

Building X 1 1 (abcd) from (33), we have g(r i )T 1 (i) = p 1 (i) ; g(r j )T 1 (j) = p 1 (j) .

Introducing the one-electron reduced matrix element

a p 1 c = -i κ a C 1 κ c V (n a κ a , n c κ c ), (36)
where V (nκ, n κ ) is the Vinti radial integral

V (nκ, n κ ) = ∞ 0 P nκ (r) d dr - κ(κ + 1) -κ (κ + 1) 2r P n κ (r)dr (37) + ∞ 0 Q nκ (r) d dr - -κ(-κ + 1) + κ (-κ + 1) 2r Q n κ (r)dr ,
the first contribution to the effective interaction strength writes as

X 1 1 (abcd) = κ a C 1 κ c κ b C 1 κ d V (n a κ a , n c κ c )V (n b κ b , n d κ d ) , (38)
recovering the uncorrected relativistic expression used in [5,6].

2.2.2. Second part: X 1 2 (abcd)
For the second term of the SMS operator, we identify from ( 6), ( 8), ( 33) and ( 34)

g(r i )T 1 (i) = -αZ r σ x ⊗ σ 1 (i) ; g(r j )T 1 (j) = p 1 (j) . (39)
Introducing the matrix element

a|r -1 σ x ⊗ σ 1 q |c = i ∞ 0 dr r -Q a P c -κ a m a |σ 1 q |κ c m c + Q c P a κ a m a |σ 1 q | -κ c m c , (40)
and using the r.m.e. (36), the corresponding contribution to the effective interaction strength is

X 1 2 (abcd) = -κ b C 1 κ d V (n b κ b , n d κ d ) × ∞ 0 dr -αZ r -Q a P c -κ a σ 1 κ c + Q c P a κ a σ 1 -κ c . (41) 2.2.3. Third part: X 1 3 (abcd)
Similarly, the two components of the third term of the SMS operator are

g(r i )T 1 (i) = -αZ r σ x ⊗ (σ r (i)C 1 (i)) ; g(r j )T 1 (j) = p 1 (j) . ( 42
)
Using the matrix element

a|r -1 σ x ⊗ (σ r C 1 q ) |c = i ∞ 0 dr r Q a P c -κ a m a |C 1 q | -κ c m c -Q c P a κ a m a |C 1 q |κ c m c , (43)
and the r.m.e. (36), the third contribution to the effective interaction strength takes the form

X 1 3 (abcd) = -κ b C 1 κ d κ a C 1 κ c V (n b κ b , n d κ d ) × ∞ 0 dr -αZ r (Q a P c -Q c P a ) , (44)
where we take advantage of

-κ a C k -κ c = κ a C k κ c .

Useful one-electron reduced matrix elements

Equations ( 26), ( 38), ( 41) and (44) are the final key expressions of the relativistic mass shift one-electron r.m.e. that involve the following three reduced angular one-electron matrix elements

κ a C 1 κ c = (-1) ja+1/2 [j a , j c ] j a 1 j c 1/2 0 -1/2 π (l a , l c , 1) , (45) -κ a σ 1 κ c = δ( l a , l c )(-1) la+1/2+ja+1 6[j a , j c ] 1/2 j a l a j c 1/2 1 ,(46) κ a σ 1 -κ c = δ(l a , l c )(-1) la+1/2+ja+1 6[j a , j c ] 1/2 j a l a j c 1/2 1 , (47)
where π (l a , l c , 1) is defined by :

π (l a , l c , 1) = 1 if l a + 1 + l c even, 0 otherwise. ( 48
)

Applications

We wrote a new program, hereafter referred as rms2, for estimating the expectation values of the relativistic nuclear recoil operators using MCDHF wave functions calculated with the grasp2K package [10]. This code is based on the previous program sms92 [6] in which

• for the NMS, the one-electron radial integrals (expression (39) of the original paper [6]) are replaced by the corresponding relativistic expression (26),

• for the SMS, the first contribution X 1 1 (abcd) (expression (40) of the original paper [6]) is corrected by adding the relativistic contributions ( 41) and (44).

It is important to notice that the program sms92 calculates the uncorrected NMS as the expectation value i T i , where T i is the Dirac kinetic energy operator

T i = c α i • p i + (β i -1 ) c 2 associated to electron i, while the program rms2 uses more accurately H 1 NMS = i p 2 i /2M
, which is consistent with section 2.1. An equivalent version has been written for the code mcdf-gme.

In the present work, we evaluate the NMS and SMS parameters ( 9) and ( 27) for some low-lying levels of neutral lithium, boron-like argon and two medium-Z carbon-like ions (Ca XV and Sc XVI) to investigate the importance of the relativistic corrections. The nuclear charge distribution is described by a Fermi model. Nuclear masses (M N ) are calculated by taking away the mass of the electrons and the binding energy from the atomic mass (M A ), using the formula:

M N (A, Z) = M A (A, Z) -Zm e + B el (Z) ( 49
)
where the total binding energy of the electrons (expressed in eV) is estimated using [START_REF] Huang | [END_REF]19] B el (Z) = 14.4381 Z 2.39 + 1.55468

• 10 -6 Z 5.35 . ( 50
)
The atomic and nuclear masses relevant to the present work are reported in Tables 1. When discussing a transition mass isotope shift, one needs to consider the variation of the mass parameter from one level to another. The line k frequency isotope shift,

δν A 1 ,A 2 k = (δE A 1 ,A 2 j -δE A 1 ,A 2 i
)/h, between the isotopes A 1 and A 2 , of nuclear masses M 1 and M 2 respectively, is usually written as the sum of the normal mass shift (NMS), specific mass shift (SMS) and field shift (FS) contributions :

δν A 1 ,A 2 k = δν A 1 ,A 2 k,NMS + δν A 1 ,A 2 k,SMS δν A 1 ,A 2 k,MS +δν A 1 ,A 2 k,FS , (51)
with

δν A 1 ,A 2 k,MS = M 2 -M 1 M 1 M 2 ∆K MS h = M 2 -M 1 M 1 M 2 ∆ K MS , (52)
where ∆K MS is the difference of the K MS parameters of the levels involved in transition k. As far as conversion factors are concerned, we use § ∆ K MS [GHz u] = 3609.4824 ∆K MS [m e E h ]. Note that thanks to the separability enhanced in (4), (52) can be applied to both the mass contributions NMS and SMS, separately.

Hydrogen-like selenium

Below we present some relevant calculations of the expression ( 26) for a heavy oneelectron ion (Se XXXIV, Z = 34). This choice is motivated by the interesting comparison with the unpublished work of Kozlov [22]. The normal mass shift values calculated with the operators H 1 NMS and (H 2 NMS + H 3 NMS ), using the rms2 program, are reported in Table 2. In the second and third column respectively, comparison is made with the numerical results of Kozlov together with our analytical values. The latter are based on analytical hydrogenic wave functions [START_REF] Johnson | Atomic Structure Theory: Lectures on Atomic Physics[END_REF] and are in complete agreement with the values of the analytical formulas derived for hydrogenic systems in [7]. Note that these formulas show that the relativistic corrections arising from the nonrelativistic and § This conversion factor is calculated as (m e /u)2R ∞ c × 1.10 -9 = 3609.4824 using the 2006 CODATA recommended values of the fundamental physical constants [START_REF] Mohr | [END_REF].

The values reported in Tables 2 and3 are based on α -1 = 137.035989500 adopted in grasp2K. For 1s 1/2 , the analytical result for K NMS becomes 656.358886872 if adopting the α -1 = 137.035999679 2006 CODATA value [START_REF] Mohr | [END_REF] .

relativistic recoil operators contain similar terms which, being rapidly growing when Z increases, cancel each other as discussed in [START_REF] Artemyev | [END_REF] for Li-like ions.

Lithium-like systems using Dirac one-electron wave functions

The SMS parameters for Li-like iron (Z = 26) and selenium (Z = 34) are calculated in the single configuration approximation using three-electron wave functions built on unscreened Dirac solutions. The results are reported in Table 3 and compared with independent estimations using an adapted version of mcdf-gme [START_REF] Indelicato | Mcdfgme, a multiconfiguration dirac-fock and general matrix elements program[END_REF]25] and with the analytical results. The three sets are consistent with each other but sensitively different from Kozlov's values [22] reported in the last column of the table. Note that the comparison is somewhat unfair to Kozlov since the grid parameters used for the discrete representation of orbital wave functions have been adapted in both programs (rms2 and mcdf-gme) to achieve a better accuracy.

Neutral lithium in the MCDHF approach

The MCDHF active space method consists in writing the total wavefunction as a configuration state function expansion built on a set of active one-electron orbitals. To investigate the convergence of the property, the orbital set is systematically expanded up to n = 10, but imposing the angular restriction l max = 6 (i orbitals). The sequence of CSFs Active Spaces (AS) is resumed as follows where the (nl)-notation implies the relativistic shell structure j = l ± 1/2. The configuration space is increased progressively, by adding at each step a new layer of variational orbitals, keeping the previous ones frozen from the (n -1) calculation.

AS 0 = 1s 2 2s, AS 2 = AS 0 + {2p} , AS 3 = AS 2 + {3s,
The MCDHF expansions are based on single and double (SD) excitations from the configuration reference. Triple excitations are investigated through SDT-configuration interaction (CI) calculations. Tables 4 and5 present the evolution of the NMS and the SMS parameter, respectively. In each table both the uncorrected (K 1 MS ) and corrected (K MS ) values are reported. Comparing the SD and SDT calculations, we observe that the influence of the triple excitations reaches more than 1 % for the SMS while it is one order of magnitude smaller (0.1%) for the NMS.

In Table 6 the individual contributions to the mass shift ∆ K MS (= ∆K MS /h) parameters are reported for the 2p 4 and5. Although many robust theoretical studies on the resonance line transition isotope shifts are available (see Table 7 and discussion below), the comparison with other theoretical works presented in Table 6 is limited to the recent large-scale configuration-interaction Dirac-Fock-Sturm calculations of Kozhedub et al [26] since these authors precisely focus on the estimation of the relativistic nuclear recoil corrections. Kozhedub et al 's values are very consistent with our results: They report ∆( K 2 NMS + K 3 NMS ) = 0.33 and 0.38 GHz u for the D 1 and D 2 transitions respectively. However, the uncorrected NMS contribution and therefore, the total NMS values, sensitively differ from each other by around 1.6 GHz u. This latter discrepancy is not understood yet and clearly deserves further investigations.

The uncorrected contribution of the SMS is also compared with the non-relativistic result of Godefroid et al [27] using the multiconfiguration Hartree-Fock method. More interesting is the comparison with the recent SMS values of Kozhedub et al [26] investigating the relativistic recoil corrections and using the same NMS and SMS partition according to ( 5) and (6). As for the NMS, the relativistic corrections are in very nice agreement (they report ∆( K 2 SMS + K 3 SMS ) = 0.12 and 0.06 GHz u for the D 1 and D 2 lines ) but the uncorrected forms do differ substantially with our estimation (they report for instance for the D 1 line, ∆ K 1 SMS = -198.78, against our value of -198.164 GHz u).

The comparison with observation for the individual mass contributions is also limited. There are a few reasons for this. First, as illustrated by (51), the field shift contribution should be properly subtracted from the observed transition frequency before trying to extract the mass contribution. But this is usually the other way round that makes the theoretical calculation of mass shifts interesting: for a fewelectron atomic systems like lithium indeed, the difference between the mass contribution calculated by elaborate ab initio calculations and the observed transition IS allows to extract the change in the mean square charge radius of the nuclear charge distributions for all isotopes, as illustrated by the very recent and complete work of Nörtershäuser et al [28]. Another good reason is that once the FS "eliminated", a clean separation of the NMS and SMS contributions could be criticized, as pointed out by Palmer [4]. However, remembering that for lithium, the FS is roughly 10 4 times smaller than the MS, it is worthwhile to neglect it for trying the mass separation exercise. There is indeed one experimental work by Radziemski et al [29] discussing the NMS and SMS separation in this line but as we will observe later (see Table 7), the corresponding experimental transition IS values are not aligned with most of the other observed values. In their work, these authors separate the two mass shift contributions from the experimental transition IS in 6,7 Li, neglecting the field shift contribution and approximating the Bohr mass shift by the experimental observed level energy, as suggested by Mårtensson and Salomonson [30],

∆ BMS = - m e M E B M - m e M E exp . ( 53
)
From the same expression, we build the transition Bohr mass shift for the 6,7 Li isotope pair

δE BMS m e M ( 6 Li) - m e M ( 7 Li) ∆E exp (54)
from the obserbed transition energy. Combining ( 52) and (54), one finds

∆ K NMS m e ∆E exp h = m e ν exp ( 55
)
from which we estimate the "observed" NMS values reported in Table 6, using the most recent absolute frequency measurements of Das and Natarajan [31]. The corresponding "observed" ∆ K SMS values are calculated by substracting the so-estimated NMS contribution from the experimental IS line shifts (-443.9490( 16) GHz u and -443.9126(29) GHz u, for D 1 and D 2 , respectively). Note that we did not take the liberty of reporting the frequency uncertainties estimated by Das and Natarajan on the separate contributions, the separability of NMS and SMS being by itself questionable. Cleaner and in principle less problematic should be the comparison of the total mass shifts, as reported in Table 7. On the theoretical side, we refer to the study of Korol and Kozlov [32] treating electron correlation with configuration interaction (CI) and many-body perturbation theory (MBPT) methods with Dirac-Fock orbitals, to the calculations of Kozhedub et al [26] using large-scale configuration-interaction Dirac-Fock-Sturm method and to the Yan et al [33] calculations estimating the mass corrections from highly correlated non-relativistic wave functions expressed in Hylleraas coordinates ¶. From all these elaborate results, we only kept the mass contributions, systematically excluding the contributions from the nuclear size corrections. We already noticed the differences between Kozhedub et al 's results and ours appearing in the separate NMS and SMS contributions. As commented above, these differences do not arise from the relativistic corrections (K 2 + K 3 ), but rather from the "uncorrected"K 1 values, and should be further investigated. Our results seem to be of higher quality than ¶ The values of Yan et al reported in the Kozhedub's paper [26] suggest that the atomic mass has been used in order to evaluate the mass shift parameter. In table 7, the Yan et al 's values have been reevaluated using the nuclear mass.

the CI+MBPT results of Korol and Kozlov. As far as the differences with Yan et al 's results are concerned, we should keep in mind i) that our orbital active set is truncated to l max = 6, ii) that the layer approach adopted in the SD-MCDHF optimization could be a limiting factor and iii) that the convergence of the ∆ K MS parameter as a function of the size of the active set is slow and not yet achieved at n = 10, as illustrated by the comparison of the two n = 9 and n = 10 sets of results reported in the Table 7.

On the experimental side, we display in the same Table 7, the experimental isotope shift values somewhat abusively converted in ∆ K MS parameters, ie. neglecting the FS contribution and inverting (52), ∆ K MS = δν k (M 1 M 2 )/(M 2 -M 1 ). As already mentioned, this conversion is unfair to physicists who do some huge efforts to extract the nuclear charge radii from the FS [28], but has the merit of illustrating where the present modest contribution lies in the distribution of experimental values. From this not exhaustive chronological list ([34, 29, 35, 36, 37, 38, 31]), it is clear that Radziemski et al 's results lie a bit outside the experimental distribution. a inverting (52), ie. using ∆ K MS = δν(M 1 M 2 )/(M 2 -M 1 ) (see text)

B-like argon

Large-scale calculations are performed for 1s 2 2s 2 2p 2 P o 1/2,3/2 of B-like argon (Z = 18). The radial orbital basis is obtained from SD-MCDHF calculations, including single and double excitations from all shells of the {1s 2 2s 2 2p, 1s 2 2p 3 } complex to increasing orbital active sets, up to the {10s9p8d7f 6g3h1i}. Subsequently to this layer-by-layer SD-MCDHF orbital optimization, RCI calculations are performed including the Breit and QED effects in a space generated by SD excitations from the extended {1s 2 2s 2 2p, 1s 2 2p 3 , 1s 2 2s2p3d, 1s 2 2p3d 2 } multireference set to the full orbital set. The expansion for the two J values includes more than 200 000 relativistic CSFs. This computational strategy has been developed by Rynkun et al [START_REF] Rynkun | At. Data Nucl. Data Tables[END_REF] for the evaluation of transition rates in boron-like ions, from N III to Zn XXVI.

Table 8 illustrates the convergence of the NMS and SMS contributions with the increasing of the active set. In Table 9, the isotope shifts of the forbidden transitions 36,40 Ar are presented and compared with the mass shift results of Tupitsyn et al [8]. In their work, the CI Dirac-Fock method was used to solve the Dirac-Coulomb-Breit equation and to calculate the energies and the isotope shifts. The CSFs expansions were generated including "all single and double excitations and some part of triple excitations". The nuclear charge distribution is described by a Fermi model and is therefore consistent with the present work. For the purpose of our mass study, we deduced the field shift from these and the Tupitsyn calculations.

Table 9 shows the individual contributions of operators ( 5) and ( 6) to the wavenumber mass shift. A good agreement is observed between the two sets of values, the total wavenumber mass shift values differing by less than 0.8%. This example beautifully confirms the importance of the relativistic corrections to the recoil operator: the total wavenumber mass shift would be indeed 50% smaller if estimated from the uncorrected form of the mass Hamiltonian H 1 NMS + H 1 SMS . Orts et al [START_REF] Soria Orts | [END_REF] succeeded to detect experimentally the IS of the transition with high precision and found a wonderful agreement with their theoretical predictions. The corresponding results are reported in Table 10 and compared with our values. The nice agreement is a good sign of reliability for the tensorial form derivation of the nuclear recoil Hamiltonian of section 2.

C-like ions calculations

As another illustration of the importance of the relativistic corrections to the recoil operator, the values of the SMS, NMS and total level mass shift parameters are reported in Table 11 for the levels arising from the ground configuration 1s 2 2s 2 2p 2 in Ca XV and Sc XVI. As far as the calculations are concerned, the orbitals are obtained by SD-MCDHF calculations, considering single and double excitations from all shells of the {1s 2 2s 2 2p 2 , 1s 2 2p 4 } Layzer complex to the {8s7p6d5f 4g2h} active set. These MCDHF calculations are followed by relativistic configuration interaction (RCI) calculations, including the Breit interaction and the QED corrections, using the enlarged multireference {1s 2 2s 2 2p 2 , 1s 2 2p 4 , 1s 2 2s2p 2 3d, 1s 2 2s 2 3d 2 } set. The size of the expansions is around 350 000 relativistic CSFs. This computational method has been used by Jönsson et al [START_REF] Jönsson | At. Data Nucl. Data Tables[END_REF] to calculate transition rates, hyperfine structures and Landé g factors for all carbon-like ions between F IV and Ni XXIII.

On the absolute scale of level shift parameters, one observes that the relativistic corrections (K 2 NMS + K 3 NMS ) to the NMS have the same order of magnitude than the uncorrected SMS contribution. Transition isotope shifts are more interesting properties since they are the real observables if the resolution is good enough. These are monitored by the differential effects on the level IS. It is interesting to infer from Table 11 the possible mass isotope shifts on the intraconfiguration (M1/E2) transition frequencies. Considering for example the Ca XV 3 P 1 → 3 P 2 transition, the uncorrected total mass shift change is enlarged by a factor two when including the (K 2 MS + K 3 MS ) relativistic corrections. For the 3 P 1 → 1 D 2 transition, a similar increase of the mass shift is predicted but of "only" 20%. Some reduction could occur: this is the case of 3 P 2 → 1 S 0 (13%). For the 3 P 0 → 1 D 2 transition, the relativistic recoil corrections reach 48%.

Conclusion and outlook

The irreducible tensorial form of the nuclear recoil Hamiltonian is derived in the present work, opening interesting perspectives for calculating isotope shifts in the multiconfiguration Dirac-Hartree-Fock framework. We implemented the formalism in the relativistic package grasp2K by writing a dedicated code (rms2) for estimating the expectation values of the relativistic nuclear recoil operators. The comparison with other works is satisfactory and the results are promising, although not achieving the accuracy of the state-of-the-art methodology available for a few-electron systems. Electron correlation remains the major problem that might be solved in our schema with the use of "localized pair-correlation functions interaction method", as proposed by Verdebout et al [START_REF] Verdebout | [END_REF]. The present work enhances the fact that the relativistic corrections to the nuclear recoil are definitively necessary for getting reliable isotope shift calculations. The new computational tool, that we developed on the basis of the irreducible tensorial operator techniques, will hopefully provide valuable mass isotope shift data for large systems for which there are reliable theoretical or experimental

  10) ‡ The two definitions of r.m.e are related to each other through γJ O k γJ = √ 2J + 1 [γJ O γJ].

Table 1 .

 1 Atomic masses (M A ) [20] and nuclear masses (M N ) (in u) calculated from (49) and (50) for lithium and argon isotopes.

	Isotope	M A	M N
	6 Li	6.015122795(16)	6.01386737
	7 Li	7.01600455(8)	7.01474907
	36 Ar	35.967545106(29) 35.9576862
	40 Ar	39.9623831225(29) 39.9525242

Table 2 .

 2 Contributions to the normal mass shift K NMS parameters (in m e E h ) for hydrogen-like selenium (Z = 34).

		rms2	Kozlov [22]	Analytic
			K 1 NMS
	1s 1/2	656.3589797	656.3589	656.358899684
	2p 1/2	154.8937900	154.8938	154.893789883
	2p 3/2	147.5192507	147.5192	147.519250700
			K 2 NMS + K 3 NMS
	1s 1/2	-78.3588949	-78.3589 -78.3588996839
	2p 1/2	-8.0987869	-8.0988	-8.0987868710
	2p 3/2	-3.0192507	-3.0193	-3.0192506997

Table 3 .

 3 Contributions to the specific mass shift K SMS (in m e E h ) parameters for Li-like iron (Z = 26) and selenium (Z = 34) using unscreened Dirac one-electron wave functions.

			rms2	mcdf-gme	Analytic Kozlov [22]
	Li-like iron			K 1 SMS
	1s 2 2p 1/2	2 P o 1/2	-55.24725067 -55.24725061 -55.247250683	-55.2474
	1s 2 2p 3/2	2 P o 3/2	-53.26443137 -53.26443136 -53.264431362	-53.2645
				K 2 SMS + K 3 SMS
	1s 2 2p 1/2	2 P o 1/2	3.48269304	3.48269307	3.482693070	3.4278
	1s 2 2p 3/2	2 P o 3/2	1.20278262	1.20278262	1.202782617	1.1960
	Li-like selenium		K 1 SMS
	1s 2 2p 1/2	2 P o 1/2	-97.71464163 -97.71464140 -97.714641685	-97.7150
	1s 2 2p 3/2	2 P o 3/2	-91.70637651 -91.70637651 -91.706376511	-91.7069
				K 2 SMS + K 3 SMS
	1s 2 2p 1/2	2 P o 1/2	10.53884731	10.5388474	10.53884746	10.2546
	1s 2 2p 3/2	2 P o 3/2	3.55081371	3.5508137	3.55081372	3.5164

Table 4 .

 4 Uncorrected (K 1 NMS ) and corrected (K NMS ) normal mass shift parameters (in m e E h ) for Li I.

	AS n	SD		SDT
		K 1 NMS	K NMS	K 1 NMS	K NMS
		1s 2 2s 2 S 1/2
	n=5	7.479955285 7.473188966	7.480387512 7.473620714
	n=6	7.480757179 7.473989593	7.481294538 7.474526401
	n=7	7.480843823 7.474076167	7.481413156 7.474644913
	n=8	7.482617678 7.475849092	7.483709525 7.476940080
	n=9	7.482764972 7.475996085	7.483865298 7.477095534
	n=10 7.482767804 7.475998981	7.483872626 7.477102933
		1s 2 2p 2 P o 1/2
	n=5	7.411878601 7.405201316	7.412125687 7.405448151
	n=6	7.412307495 7.405629843	7.412599631 7.405921698
	n=7	7.412593434 7.405916172	7.413034981 7.406357367
	n=8	7.414193990 7.407516244	7.415203718 7.408525555
	n=9	7.414351543 7.407673608	7.415377292 7.408698936
	n=10 7.414365017 7.407687009	7.415402512 7.408724081
		1s 2 2p 2 P o 3/2
	n=5	7.411871260 7.405208436	7.412118064 7.405455006
	n=6	7.412300503 7.405637317	7.412592555 7.405929108
	n=7	7.412584979 7.405922413	7.413026010 7.406363146
	n=8	7.414185599 7.407522728	7.415193271 7.408530126
	n=9	7.414343399 7.407680348	7.415366987 7.408703663
	n=10 7.414356793 7.407693666	7.415392260 7.408728857

  S 1/2 (D 2 line) transitions in lithium. Values are calculated with the SD and SDT n = 10 active space final results of Tables

	1/2	2 P o 1/2 -2s 2 S 1/2 (D 1 line) and 2p 3/2	2 P o 3/2 -
	2s 2

Table 5 .

 5 Uncorrected (K 1 SMS ) and corrected (K SMS ) specific mass shift parameters (in m e E h ) for Li I.

	AS n	SD		SDT
		K 1 SMS	K SMS	K 1 SMS	K SMS
		1s 2 2s 2 S 1/2
	n=5	0.3010343291 0.3008225633	0.3013767853 0.3011648528
	n=6	0.3010361585 0.3008243666	0.3014579841 0.3012459847
	n=7	0.3019544951 0.3017423943	0.3024396569 0.3022273237
	n=8	0.3018617523 0.3016497153	0.3024115843 0.3021992791
	n=9	0.3017987398 0.3015867742	0.3023512554 0.3021390203
	n=10 0.3018561821 0.3016442119	0.3024141615 0.3022019200
		1s 2 2p 2 P o 1/2
	n=5	0.2489342617 0.2487564343	0.2490604867 0.2488826064
	n=6	0.2482881614 0.2481107089	0.2484282378 0.2482507247
	n=7	0.2482993836 0.2481222326	0.2484107486 0.2482335819
	n=8	0.2476015693 0.2474248242	0.2474557225 0.2472791143
	n=9	0.2475207029 0.2473440589	0.2473719731 0.2471954806
	n=10 0.2476566450 0.2474799659	0.2475131224 0.2473365865
		1s 2 2p 2 P o 3/2
	n=5	0.2489331884 0.2487377216	0.2490592224 0.2488636586
	n=6	0.2482892038 0.2480941432	0.2484289962 0.2482338261
	n=7	0.2483022860 0.2481070388	0.2484139538 0.2482185782
	n=8	0.2476039924 0.2474089620	0.2474585594 0.2472634414
	n=9	0.2475233735 0.2473284034	0.2473750253 0.2471799634
	n=10 0.2476606363 0.2474656360	0.2475176569 0.2473225644

Table 6 .

 6 Individual contributions to the mass shift ∆ K MS (GHz u) parameters for the 2p 2 P o 1/2 -2s 2 S 1/2 and 2p 2 P o 3/2 -2s 2 S 1/2 transitions in lithium

	2 P o 1/2 -2 S 1/2	2 P o 3/2 -2 S 1/2

a CI Dirac-Fock-Sturm calculation of Kozhedub et al

[26]

. b NMS values deduced from the transition frequencies

[31]

using (55) (see text). c Non-relativistic MCHF calculations

[27]

. d SMS values obtained by subtracting the "observed" NMS (see footnote b above) from the IS measured by Das and Natarajan

[31]

.

Table 7 .

 7 Mass shift ∆ K MS (GHz u) for the 2p 1/2 2 P o 1/2 -2s 2 S 1/2 and 2p 3/2 2 P o 3/2 -2s 2 S 1/2 transitions in lithium, compared with experimental IS.

		2 P o 1/2 -2 S 1/2	2 P o 3/2 -2 S 1/2	Ref.
	n = 9	-445.1941	-445.2330	This work
	n = 10	-444.8442	-444.8808	This work
	Other theory -447(12)	-447(12)	Korol and Kozlov [32]
		-443.81(20)	-443.82(20)	Kozhedub et al [26]
		-443.860337(253) -443.876984(253) Yan et al [33]
	Experiment a -443.89(3)	-443.91(2)	Sansonetti et al [34]
		-443.46(63)	-443.59(63)	Radziemski et al [29]
		-443.9033(63)	-443.9791(63)	Scherf et al [35]
		-443.9045(29)		Bushawet al [36]
		-443.951(5)		Walls et al [37]
		-443.941(3)	-443.948(4)	Noble et al [38]
		-443.9490(16)	-443.9126(29)	Das and Natarajan [31]

Table 8 .

 8 NMS and SMS parameters (in m e E h ) values for the states 1s 2 2s 2 2p 2 P o

	1/2

Table 9 .

 9 Individual contributions to the wavenumber mass shift δσ (cm -1 ) for the forbidden transition 1s 2 2s 2 2p 2 P o 1/2 -2 P o 3/2 in boron-like36,40 Ar.

			δσ
		H 1 NMS	H 2 NMS + H 3 NMS	H 1 SMS	H 2 SMS + H 3 SMS	Total
	This work	0.1054	-0.0821	-0.0745	0.1155	0.0644
	Tupitsyn et al [8]	0.1053	-0.0822	-0.0742	0.1151	0.0640

Table 10 .

 10 Wavelength mass shift δλ (nm, air) for the forbidden transition 1s 2 2s 2 2p 2 P o 1/2 -2 P o 3/2 in boron-like36,40 Ar.

		λ ( 40 Ar)	δλ ( 36,40 Ar)
	This work	441.01	0.00125
	Orts et al [40] th.	441.16(27)	0.00123(5)
	obs. 441.2556(1) 0.00123(6)

Table 11 .

 11 Specific mass shift K SMS , normal mass shift K NMS , total mass shifts K MS parameters (all in m e E h ) for 2s 2 2p 2 levels of Ca XV and Sc XVI from multireference RCI calculations. K = K 1 + K 2 + K 3 .

				SMS	NMS		Total
	Level	J	K 1 SMS	K SMS	K 1 NMS	K NMS	K 1 MS	K MS	K 2 MS + K 3 MS
					Ca XV
	2s 2 2p 2 3 P 0 -41.993 -40.659	558.358 538.033	516.364 497.374	-18.990
		1 -41.736 -40.741	558.070 537.983	516.334 497.242	-19.092
		2 -41.588 -40.821	557.854 537.929	516.266 497.108	-19.158
	2s 2 2p 2 1 D 2 -41.451 -40.756	557.479 537.607	516.028 496.851	-19.177
	2s 2 2p 2 1 S	0 -41.862 -41.199	557.000 537.200	515.138 496.001	-19.137
					Sc XVI
	2s 2 2p 2 3 P 0 -47.460 -45.771	621.221 596.303	573.761 550.532	-23.229
		1 -47.117 -45.874	620.842 596.236	573.725 550.362	-23.363
		2 -46.950 -45.964	620.592 596.169	573.642 549.219	-24.423
	2s 2 2p 2 1 D 2 -46.765 -45.922	620.145 595.826	573.380 549.904	-23.476
	2s 2 2p 2 1 S	0 -47.229 -46.421	619.624 595.390	572.395 548.969	-23.426
	values.

Acknowledgements

Cédric Nazé is grateful to the "Fonds pour la formation à la Recherche dans l'Industrie et dans l'Agriculture" of Belgium for a Ph.D. grant (Boursier F.R.S.-FNRS). Michel Godefroid and Cédric Nazé thank the Communauté française of Belgium (Action de Recherche Concertée) for financial support. MRG also acknowledges the Belgian National Fund for Scientific Research (FRFC/IISN Convention). Finally, the authors thank Jiguang Li for fruitful discussions.


Gábor J Halász

Ágnes Vibók

Milan Šindelka

Nimrod Moiseyev

S Lorenz

Cederbaum

Lorenz S Cederbaum

Conical intersections induced by light: Berry phase and wavepacket dynamics

published or not. The documents may come   L'archive ouverte pluridisciplinaire

I. Introduction

Conical intersections (CIs) between electronic potential energy surfaces play a key mechanistic role in the nonadiabatic molecular processes [13]. In this case the nuclear and electronic motion can couple and the energy exchange between the electrons and nuclei may become signicant. In several important cases like dissociation, proton transfer, isomerization processes of polyatomic molecules or radiationless deactivation of the excited state systems [49] the CIs can provide very ecient channel for ultrafast interstate crossing on the femtosecond time scale. CIs can be evolved between dierent electronic states starting from triatomic systems to a truly large polyatomic molecules. Several important books, review articles and publications have demonstrated the existence and relevance of such intersections in recent decades [START_REF] Worth | [END_REF]1012].

Generally CIs are not isolated points in the conguration space. Rather, they are an innite number of connected points forming a seam. For a diatomic molecule that has only one degree of freedom, it is not possible for two electronic states of the same symmetry to become degenerate and as a consequence of the well-known noncrossing rule an avoided crossing results. It should be noted, however, that this statement is true only in free space. As it was shown in earlier papers [13,14] CIs can be formed both by standing and running laser waves even in diatomic systems. In the former case the laser light-induces CIs which couple the center of mass motion with the internal rovibrational degrees of freedom. In the latter case, the rotational motion constitutes the missing degree of freedom to allow for the formation of a CI. It was found that in a diatomic system under the inuence of a standing laser eld (optical lattice) a periodic array of CIs is formed by the laser light [13]. In addition, it was shown that the appearance of these light-induced CIs can signicantly reduce the magnitude of the trapping eect of cold diatomic molecules in the lowest electronic state.

For the case of running laser waves, it was demonstrated that there is a very strong impact of the light-induced CIs on the molecular spectrum and spectral properties [14].

We would like to mention that often the rotational degree of freedom is neglected and only the vibrational one is taken into account in computing diatomic molecules in laser elds, and thus the CI is neglected (see e.g. [15 17] and citations therein). Previous works demonstrated, however, the importance of the rotations in the presence of an intense laser eld [18,19].

The light-induced CIs constitute a novel and physically interesting new laser-matter phenomenon. The presence of these light-induced CIs in diatomic systems may completely change their original, i.e., eld free, physical properties. In other words, using either standing or running laser waves it is possible to build in signicant nonadiabatic eects into molecular systems using manipulable external forces allowing one to open up a new direction in the eld of molecular quantum control processes.

The light-induced strong nonadiabaticity couples in a controllable way dierent electronic states of the molecule and depending on the eld intensity can be extremely large in the vicinity of the CIs. The nonadiabatic couplings become singular at the CIs [START_REF] Köppel | [END_REF]. The presence of a CI provides the source for numerous statical and dynamical nonadiabatic phenomena. One of these phenomena is the appearance of a topological eect often referred to as Longuet-Higgins or Berry phase [2123]. In this work as a continuation of the former ones [13,14] we choose again the Na 2 molecule as a sample system. One of our aims is to calculate the topological or Berry phase in Na 2 . The presence of this phase is a clear ngerprint of the laser-induced CI.

As well studied in free polyatomic molecules, the dynamics of the system is highly aected by CIs and becomes strongly nonadiabatic owing to the strong coupling between the nuclei and electrons. The importance of this feature cannot be overemphasized. Therefore, investigating the dynamics in the region of laser induced CI is another goal of this paper. The presented wave packet propagation study is expected to be of particular interest and to reect the eect of the light-induced CI.

The paper is organized as follows. Section II gives a brief description of our working Hamiltonian. In Section III a short characterization of the CIs and of the resulting nonadiabatic couplings is presented. We present the Berry phase and the results of the wave-packet calculations related to the two lowest states of sodium dimer in Section IV and V, respectively. These sections are also devoted to the discussion. Conclusions are presented in Section VI.

II. The Hamiltonian

The results reported here are obtained by solving the time-dependent Schrödinger equation for the nuclear motion of Na 2 proceeding on two coupled potential energy surfaces. The respective electronic states are coupled by a running laser wave of wavelength λ, and labeled X (ground, X 1 Σ + g ) and A (excited,

A 1 Σ + u ): i d dt ψ X ψ A = Ĥ ψ X ψ A , (1)
where the total molecule and eld Hamiltonian is Ĥ = ( Ĥvib + Ĥrot ) + Ŵ.

(2)

Here, Ĥvib and Ĥrot are the molecular vibrational and rotational kinetic energy Hamiltonian, respectively, and Ŵ is the potential energy matrix in the above two electronic states. These terms are given by

Ĥvib = -1 2 2µ ∂ 2 ∂R 2 , (3)
Ĥrot = 1 L 2 θϕ 2µR 2 , (4)
µ being the reduced mass of the diatomic molecule, R and (θ,ϕ) are its vibrational and rotational coordinates, respectively, and L 2 θϕ is the angular momentum opera- tor of the nuclei. The molecule-light term coupling the two electronic states can be written as the scalar product of the transition dipole moment -→ d and the electric-eld vector -→ ε :

- → d • - → ε = 0 d(R) cos θ cos ω L t. ( 5
)
Here 0 is the maximum laser eld amplitude, ω L is the laser frequency which couples the two electronic states, d(R) is the transition dipole matrix element. For symmetry reason only the z component of the transition dipole moment (d(R) =< ψ e 1 | j z j |ψ e 2 >) is non-vanishing. The potential part Ŵ contains the eld-free potential energy curves V X and V A of the two electronic states as diagonal elements and the radiative coupling as the nondiagonal element. In the static dressed state representation this potential energy matrix has the following form:

Ŵ = V X ( 0 /2)d(R) cos θ ( 0 /2)d(R) cos θ V A -ω L . (6)
The potential energies V X and V A of the two electronic states and the transition dipole moment were taken from [24].

III. Conical Intersection and Nonadiabatic Coupling Matrix

In eq. ( 6) of Sec. II. we gave the expression for the potential energy matrix Ŵ . To continue, we dene as usual the unitary matrix Û which diagonalizes the potential energy matrix and thus transforms the Hamiltonian [START_REF] Baer | Beyond Born Oppenheimer: Electronic Non-Adiabatic Coupling Terms and Conical Intersections[END_REF] to its adiabatic representation:

U (R, θ) = cos Φ(R, θ) sin Φ(R, θ) -sin Φ(R, θ) cos Φ(R, θ) , (7)
with

Φ(R, θ) = 1 2 arctan ε 0 d(R) cos θ V A (R) -ω L -V X (R) . (8)
In this picture the Û Ŵ Û † transformation produces the adiabatic potential energy surfaces, V lower ad (R, θ) and V upper ad (R, θ). These two potential energy surfaces can cross each other, forming a real conical intersection (see Figure 1), only if the two conditions (cos θ = 0, (θ = π/2) and V X (R) = V A (R) -ω L ) are simultaneously fullled.

The characteristic features of this CI can be changed by varying the frequency and intensity of the light eld. Increasing the frequency, for example, moves the CI to a smaller internuclear distance and to a smaller energetic position. While the steepness of the CI cone formed by the adiabatic surfaces, which is related to the strength of the nonadiabatic coupling, can be controlled by the laser intensity.

The molecule-light couplings are eliminated from the potential energy matrix by applying the diagonalizing transformation specied above.

On the other hand, nonadiabatic couplings now appear in the transformed kinetic energy operator. As is known from the literature

In Figure 2 we display the nonadiabatic couplings in the close vicinity of the light-induced CI. As expected their values can be very large and diverge at the CI as is the case for natural CIs of polyatomics (compare with [START_REF] Köppel | [END_REF]).

This leads us to conclude that the light-induced CIs introduce similar nonadiabatic eects as in polyatomics such that the usual Born-Oppenheimer approximation will not be valid any more.

IV. Topological or Berry Phase

Having discussed some important features of the lightinduced CIs, let us study now the impact of this intersection on physical properties of the Na 2 . To continue we turn to the topological or Berry phase.

It was rst pointed out by Longuet-Higgins and Herzberg [21,22] that each real adiabatic electronic state changes sign when transported continuously along a closed loop enclosing the point of conical intersection.

Mead and Truhlar connected this geometric phase eect with the single electronic state problem [25] and Berry generalized the theory [23]. Hence the name Berry phase.

As the total wave function must be single valued, it is favorable to make the electronic wave function in the adiabatic representation complex by multiplying it by a phase factor ensuring that the total wave function remains single valued. The fact that the electronic eigenfunctions are modied has a direct eect on the nuclear dynamics even when a single potential energy surface is considered. Consequently, the appearance of the Berry phase in a molecular system can be considered as a clear signature of the conical intersection independently of whether it is a natural or a laser-induced one.

To continue, we refer to the literature [START_REF] Baer | Beyond Born Oppenheimer: Electronic Non-Adiabatic Coupling Terms and Conical Intersections[END_REF] that the topological or Berry phase α 12 can be calculated for a closed contour Γ as:

α 12 = Γ τ 12 (s ) • ds , (10)
where τ 12 is the non-adiabatic coupling term between the two electronic states, the components of which are introduced in the previous section. As a consequence of sign changes of the adiabatic states, it can be proven that the value of this phase is equal to:

α 12 = π × 2n + 1 if Γ encircles odd number of CIs 2n if Γ encircles even number of CIs n = 0, ±1, ±2, ... (11)
Figure 2:

The nonadiabatic coupling terms (τ12R = ∂Φ/∂R and τ 12θ = ∂Φ/∂θ) for Na2. The applied eld intensity and photon energy are: I = 3.0×10 10 W cm 2 and ωL = 1.968 eV. Now, using eq. ( 9), it is possible to calculate the Berry phase α 12 for a closed path Γ as follows:

α 12 = Φ(s0) end of the path -Φ(s0) begining of the path . (12) In Fig. 3a we display the energy curve V X of the elec-

V. Wave Packet Dynamics

In this section we discuss the impact of the lightinduced CI on the time-dependent wave packet propagation. We demonstrate that the appearance of lightinduced conical intersections leads to substantial and non-trivial eects in the dynamics. As is well known from the studies of conical intersections in polyatomics, the details of the dynamics are highly intricate and cannot be understood without explicit computations. It is in The results presented here are obtained by solving numerically the time-dependent Schrödinger equation [START_REF]Conical Intersections: Electronic Structure, Dynamics and Spectroscopy[END_REF].

The evolution of the wave function is calculated by using the usual time propagator

|ψ(R, θ, t) >= exp(- i H t)|ψ(R, θ, 0) >, (14)
where the time-independent Hamiltonian Ĥ is given by eqs.( 2) and ( 6).

We expressed this Hamilto-nian in matrix representation in the basis set of the free molecular electronic and rovibrational states (|ϕ X ν,J (R, θ) >; 0) T and (0; |ϕ A ν,J (R, θ) >) T . The spatial components of these free molecular states were calculated using the particle in a box functions, 2/L sin(kπR/L) k=1,2,...,N b which are multiplied by Legendre polynomials {P j (cos θ)} j=0,1,2,...,N J . In our nu- merical calculations we have used N b = 500 and N J = 95

for the X surface whereas for the A surface we have considered correspondingly 96 Legendre polynomials. In this representation the eigenvalues (E j ) and eigen functions (φ j ) of the Hamiltonian can be obtained numerically.

Next, one can describe the initial wave function in terms of these eigen functions: |ψ(R, θ, t = 0) = j c j φ j .

Based on this partitioning of the initial wave function one can reformulate equation 14:

|ψ(R, θ, t > 0) = j exp - i E j t c j φ j . ( 15
)
Throughout the paper the initial wave function is chosen as the electronic and rovibrational ground state solution of the eld free Hamiltonian:

|ψ(R, θ, t = 0) = ϕ X ν=0, J=0 (R, θ) ; 0 T .
To shed more light on the meaning of the wave packet propagation in the present context we stress that we are calculating the dynamics of a system which is rst in a eld free space and then suddenly exposed to the running laser wave. It is shown in the Appendix that this physical situation amounts to propagating the initial eld free molecular state with the Hamiltonian [START_REF] Baer | Beyond Born Oppenheimer: Electronic Non-Adiabatic Coupling Terms and Conical Intersections[END_REF][START_REF] Worth | [END_REF][START_REF] Klessinger | Excited States and Photochemistry of Organic Molecules[END_REF][START_REF] Kukura | [END_REF](6). In order to compare with experiments, it suces that in the latter the time of switching on the wave at the molecular site must be much shorter than the period of the laser.

The solution of eq.( 14) is used to calculate the autocorrelation function which is the overlap between the initial and the time evolved wave packets

C(t) = | < ψ(R, θ, 0)|ψ(R, θ, t) > | (16) = π 0 dθ • sin θ ∞ 0 dR • ψ(R, θ, 0) * • ψ(R, θ, t) .
as well as the time-dependent population on dierent adiabatic or diabatic surfaces (SF )

P SF (t) = < ψ SF (R, θ, t)|ψ SF (R, θ, t) >= (17) π 0 dθ • sin θ ∞ 0 dR • ψ SF (R, θ, t) * • ψ SF (R, θ, t).
More specically, we study the following quantities We begin with the weak eld situation (I = 3.0 × 10 8 W cm 2 ).

P X dia (t), P A dia (t)
In Fig. 5 we rst display the results for ω L = 1.851 eV . Here, the energy of the initial wave packet is slightly above the bottom of the V A -ω L curve, but still below the lowest vibrational level of angular momentum J = 1 on the A surface to which it couples directly. Furthermore, the energy of initial wave packet is also below that the energy of the CI. It is known that in such cases the nonadiabatic eects on the initial wave packet are weak [START_REF] Köppel | [END_REF]. Consequently, the whole wave packet stays localized on the X surface up to 50 ps and the populations on the two diabatic surfaces remain practically unchanged. Turning to the adiabatic picture, the situation is similar. The population on the upper surface is more or less constant and very low (around ∼ 0.004) apart from the rst 1 -2 ps. On the other hand, the autocorrelation function does show clear periodical changes which due to fact that the initial eld free state is a linear combination of essentially two eigenfunctions of the full Hamiltonian [START_REF] Baer | Beyond Born Oppenheimer: Electronic Non-Adiabatic Coupling Terms and Conical Intersections[END_REF]6). Ordered by increasing energy, these are the rst and third eigenstates φ 1 and φ 3 , and are initially populated by ∼ 73.3% and ∼ 26.4%, respectively.

We now increase the laser frequency to ω L = 1.968 eV . The results of the computations are collected in Fig. 6 in analogy to Fig. 5. In this situation both the energetic and internuclear positions of the CI change to lower values. Importantly, the position of the initial wave packet is now higher by about 0.1 eV than the bottom of the V A -ω L curve, and is also above the position of the CI. In contrast to the rst example, the wave function can now spread out more eciently through the CI.

The decay of the autocorrelation function is rather fast: it arrives at its rst zero after 1.34 ps. At later times it oscillates and exhibits a larger recurrence at around 9.5 ps and thereafter continues to oscillate with a larger amplitude than before this recurrence. In this case the initial wave function is a linear combination of several eigenfunctions. Nine of them cover more than 98% of the initial wave function. The periods of times related to the energy dierences between these eigenfunctions are in the range from 1.5 ps to 20.1 ps and ve of them -between neighbouring states -are between 9.8 ps and 10.9 ps. The interference of these states causes the irregular oscillations and the recurrences of the autocorrelation function. All populations show large variations with time. The population on the X surface is changing Because of the applied weak eld in the cases studied above in Figs. 5 and6, the energy split between the adiabatic upper and lower curves at the crossing point of the diabatic curves and θ = 0 is extremely small and almost negligible (it amounts to ∼ 10 meV ). However, the picture changes markedly in the case of the higher eld intensity studied here, I = 3.0 × 10 10 W cm 2 . These adiabatic curves are depicted in addition to the diabatic curves in panels (a) of Figs. 7 and8. Increasing the eld intensity increases the energy split substantially as expected from the o diagonal term in the potential matrix Ŵ in (6). At the value of R of the CI where the respective diabatic curves cross, the energy split is twice the absolute value of this o diagonal term. In polyatomic molecules it is known that the appearance of nonadiabatic eects intensies due to conical intersections when the so-called interstate coupling constants increase [START_REF] Köppel | [END_REF].

In the present context of light-induced CIs the analogous interstate coupling constant is given by 0 d(R)/2 as can be seen by comparing Eq.( 6) and the respective equations in Ref. [START_REF] Köppel | [END_REF]. Since this interstate coupling constant increases with increasing eld intensity, we expect stronger nonadiabatic eects in stronger elds, in general.

Again, as for the weak eld, we study here two situation with dierent laser frequencies. We depict in Fig. 7 the results for the lower frequency ω L = 1.851 eV . We see that the autocorrelation function now decays very fast within the rst 200 f s to a rather low value of about ∼ 0.35 and continues to drop to about ∼ 0.21 during the next 400 f s. The population on the X surface is permanently decreasing (from 1 to ∼ 0.55) during the rst 600 f s. In contrast, the population on the upper adiabatic surface changes very little (see the scale on the right hand side of the gure). This is due to the fact that the initially populated level is well below the energy of the CI and essentially within the lower adiabatic surface, see Fig. 7(a).

The results obtained for the larger laser frequency ( ω L = 1.968 eV ) are summarized in Fig. 8. The energy of the initial wave packet is now slightly above the energy of the CI and is signicantly larger than the bottom of the V A -ω L curve. We see that due to this energetic situation the dynamics is pretty fast and the autocorrelation function drops signicantly to only about 0.23 within the short time of just 75 f s. Later on this function continues to decay exhibiting oscillations. This behavior of the autocorrelation function is rather similar to those found in small polyatomics with eld free CIs [START_REF]Conical Intersections: Electronic Structure, Dynamics and Spectroscopy[END_REF][START_REF] Worth | [END_REF]. The populations of the diabatic and in this example of the adiabatic states as well also show rapid substantial changes on a similarly short time scale. Interestingly, the population on the diabatic A surface, which is zero initially, now becomes considerably larger than that of the X surface at most of the time interval depicted in Fig. 8(b).

Finally, we would like to gain some insight into the impact of the CI by freezing the rotational degree of freedom. For this purpose we assume that the orientation of the system is xed during the process and the molecular axis is parallel to the polarization of the laser eld, i.e., we set cos θ = 1 in eq.( 6), to maximize the coupling between the electronic states. In what follows we compare the results of this calculations (referred to as the no CI case) with the above shown full calculations (CI case). The results obtained for the autocorrelation function and the adiabatic population are collected in Fig. 9 for ω L = 1.968 eV and I = 3.0 × 10 10 W cm 2 where they are also compared with the respective quantities taken from Fig. 8. The dierences between the no CI and CI results are obviously considerable. First, the behavior of the no CI autocorrelation function is rather structured, while the correct function is nearly smooth showing a more or less continuous decay. This dier-ence is due to the much larger density of states available and populated in the CI case where many odd J values contribute in contrast to only J=0 in the no CI case. Interestingly, the no CI autocorrelation function decreases much faster at very short times (up to 30 f s) than the correct one, and at later times it shows dramatic recurrences up to a value of 0.8 in sharp contrast to the decaying function in the CI case.

Another quantity of interest is the population of the upper adiabatic state. Both the CI and no CI results are rather comparable in the short time period where the autocorrelation function drops fast (about 30 f s). At longer times, the deviation between the populations become quite remarkable. In the no CI case the adiabatic population jumps back close to its original value and then falls o relatively atly except of the bump occurring at around 300 f s, a time when the autocorrelation function has its large recurrences. In contrast, the correct population of the upper adiabatic state continues to decay and at the end of the 600 f s shown in Fig. 9 more than 60% of its initial value has been transferred to the lower adiabatic state.

One may be tempted to expect that if the laser intensity is weak, say I = 3.0 × 10 8 W cm 2 , the deviations of the no CI from the CI results will be hardly visible. This is, however, not necessarily the situation. In Fig. 10 we display in analogy to Fig. 9 the results obtained for the weak intensity. The no CI results are surprisingly different from the CI ones already discussed above. First of all, we notice that like in the stronger eld (see Fig. 9), also here the autocorrelation function in the no CI case decreases more rapidly at the very beginning than in the CI case. But the most striking feature of the autocorrelation function in the no CI case is the appearance of nearly complete periodic recurrences where the wave packet returns to the initial wave function. We shall return to this behavior below. Concerning the population on the upper adiabatic surface, it seems rather meaningless to compare between the no CI and the CI cases.

The former one displays much more similarities to the autocorrelation function in the no CI case than having any resemblance to the result of the CI case.

To understand the behavior in the no CI case, we should study some details of the actual wave function. It turns out that at the given photon energy accidentally two eigenstates of the eld free Hamiltonian energy split of ∼ 3 meV leads to periodic recurrences in the autocorrelation function with a time period of about T ≈ 1.36 ps as observed in the calculations. In the CI case the situation is rather dierent as initial state having J = 0 couples directly only to eld free states of J = 1 which in turn couple to J = 0 and 3 states etc.

Consequently, the nearly degenerate state which is responsible for the striking beating found in the no CI case is irrelevant in the full calculation. To demonstrate the impact of the CI we briey mention that the starting wave function when expressed by the respective eigen-functions acquires contributions of similar weights from 8 eigenfunctions. Moreover, when expanded in eld free states, the eigenfunctions in the CI case are found to be mixtures of several rotationally excited eigenstates of the eld free problem on the two diabatic surfaces with signicant coecients (> 0.2) up to J = 13.

VI. Conclusions

Recently, a novel and physically interesting phenomenon was found [13,14] + E 0 cos(θ)d(R) cos(ω L t) •

•

[|Ψ elec A Ψ elec X | + |Ψ elec X Ψ elec A |]
where H X is the time-independent Hamiltonian with the ground electronic energy as function of R as a potential and similarly H A is the time-independent Hamiltonian for the rst excited electronic state.

The initial state is the eld-free state of the molecule which is suddenly exposed to the cw laser beam. In our case the initial state is taken as the ground vibrational and rotational eigenstate of H X , χ X ν=0,J=0 and therefore, |Ψ(t = 0) >= χ X ν=0,J=0 |Ψ elec X .

(

) 20
Using the Shirley approach [26] we can expand the solution of the TDSE with the quasi-energy (Floquet) solutions which we denote here by |Ψ QE α (t) > (we will explain what they are later). Following Shirley the solution of Eq.18 is given by,

|Ψ(t) >= α C α Ψ QE α (t) (21)
where

C α = Ψ QE α (t = 0)|Ψ(t = 0) . (22)
The quasi-energy (Floquet) solutions are dened as (see for example, [START_REF] Faisal | Theory of Multiphoton Processes[END_REF][START_REF] Moiseyev | Non-hermitian Quantum Mechanics[END_REF])

|Ψ QE α (t) = e -iE QE α t/ ∞ n=-∞ e iω L nt |ϕ n,α (23)
where correspondingly E QE α and the n-th components of the vector | -→ ϕ α > are the eigenvalues and eigen-states of the time-independent Floquet matrix, H F , which its (n, n ) matrix element is dened as Even when the laser eld is strong the Fourier components can be truncated and the Floquet matrix can be truncated. For the case where the photoinduced dynamics is a net one-photon process (as in our case) it is enough to take {n , n} = 0, -1 or {n , n} = 0, +1 and the initial state is the eld-free state which is dened above in Eq.22. For our studied case the time-independent Floquet matrix which is given above in Eq.27 and the propagation of the wavepacket within the framework of the Floquet formalism as given above on Eq.26 are as dened in Eqs.1-6 in our paper.

[H F ] n,n = (24) 1 T

Figure 1 :

 1 Figure 1: Light-induced conical intersection between the V lower ad (R, θ) and V upper ad (R, θ) adiabatic potential energy surfaces in the Na2 molecule. The applied eld intensity and photon energy are: I = 3.0 × 10 10 W cm 2 and ωL = 1.968 eV.

[ 2 ,

 2 10], these nonadiabatic couplings can be calculated as the derivatives of the transformation angle Φ with respect to the corresponding nuclear coordinate τ 12R = ∂Φ/∂R and τ 12θ = ∂Φ/∂θ.

Figure 3 :

 3 Figure 3: (a) Potential energy VX and eld-dressed potential energy VA -ωL curves of Na2. The X and A notations correspond to the X 1 Σ + g and A 1 Σ + u electronic states. (b) Geometrical arrangement of the contours used in the topological phase calculations. The angle β {0...2π} parametrizes the actual position in the conguration space where the transformation angle is calculated. The black triangle shows the position of the laser-induced conical intersection. (c) Transformation angles as a function of β for three dierent geometrical arrangements. Only the solid curve is calculated along a path surrounding the CI. The arrows denote that the solid and dashed functions are given by the scale on the left side while the value of the dotted function is given by the scale on the right side.

Figure 4 :

 4 Figure 4: Color mapped plots of the transformation angle as a functions of R and θ for two intensities of the laser eld: (a) I = 3.0 × 10 8 W cm 2 and (b) I = 3.0 × 10 10 W cm 2 .

  and P upper ad (t), which correspond to the population of (or probability of being on) the ground state diabatic surface (SF = X), on the excited state diabatic surface (SF = A) and on the adiabatic upper state surface (SF = upper)[30], respectively. In the numerical calculations the two eld intensities I = 3.0 × 10 8 W cm 2 and I = 3.0 × 10 10 W cm 2 , and the two dierent values of the photon energy 1.851 eV and 1.968 eV were used. The internuclear (R) and the energetic (E) positions of the light-induced conical intersections are easily determined (see below) and, of course, do not depend on the used laser intensity. The dynamical investigations performed can be classied according to the actual position of the conical intersections and intensity. To avoid numerous gures showing the many results obtained, we collect the numerical results for the autocorrelation and various population functions compactly in a single picture for each set of parameters (photon energy and eld intensity).

Figure 5 :

 5 Figure 5: Potential curves and results for eld intensity I = 3.0 × 10 8 W cm 2 and frequency ωL = 1.851 eV . (a) Potential energy VX and eld-dressed potential energy VA -ωL curves of Na2. The dashed-dotted vertical line indicates the geometric position of the CI and the dashed-dotted horizontal line the energy position of the initial state of the propagation. (b) Autocorrelation function (solid line), population on the ground state diabatic surface (dotted line), on the excited state diabatic surface (dashed line) and on the adiabatic upper state surface (• • •) are shown as a function of time.

Figure 6 :

 6 Figure 6: The same as Figure 5 except that the laser frequency is now increased to ωL = 1.968 eV . For more details see Fig.

5 .

 5 rapidly within the 1 st ps and then continues to oscillate around approximately the value of ∼ 0.4 also exhibiting -as in the case of the autocorrelation function -a local enhancement at around 9.5 ps. A recurrence in the autocorrelation function implies that the wave packet has returned partly to the initial one. Since the initial wave packet in here starts from the ground state X surface, the population on this surface may also grow around the recurrence. The population of the adiabatic upper state follow approximately the shape of that of the X state.

; 8 T

 8 and ϕ A ν=8 ) are nearly degenerate: E A 8 -E X 0 ≈ 25 µeV . In the laser eld these two eld free states will strongly mix giving rise to the two new eigenstates: |φ 522 ≈ 0.68 ϕ X 0 , with which have an energy split of: E 552 -E 522 ≈ 3.06 meV . Our initial wave function at t = 0 is the electronic and vibrational ground state of the eld free system (ϕ X ν=0 ), and can be excel- lently expressed by these two eigenstates: |Ψ(t = 0) = ϕ X 0 ; 0 T ≈ 0.68 |φ 522 + 0.71 |φ 552 . At later times we have a beating between these two eigenstates and the

Figure 7 :

 7 Figure 7: Potential curves and results for eld intensity I = 3.0 × 10 10 W cm 2 and frequency ωL = 1.851 eV . (a) Potential energy VX and eld-dressed potential energy VA -ωL curves of Na2. Shown are also the respective adiabatic potential curves at θ = 0 as dashed curves. The dashed-dotted vertical line indicates the geometric position of the CI and the dashed-dotted horizontal line the energy position of the initial state of the propagation. (b) Autocorrelation function (solid line), population on the ground state diabatic surface (dotted line), on the excited state diabatic surface (dashed line) and on the adiabatic upper state surface (• • •) are shown as a function of time.

Figure 8 :

 8 Figure 8: The same as Figure 7 except that the laser frequency is now increased to ωL = 1.968 eV . For more details see Fig.

Figure 9 :

 9 Figure 9: Autocorrelation function and population on the adiabatic upper state surface as a function of time for the no CI case where the rotational motion has been frozen (cos θ = 1 in eq.(6)). The results are compared with those of the full calculations (CI case). The laser eld intensity and photon energy are: I = 3.0 × 10 10 W cm 2 and ωL = 1.968 eV . The curves of the autocorrelation function are marked with (solid line) and (dotted line) for the CI and no CI cases, respectively. The curves of the upper state population are marked with (• • •) and (

  in the eld of light-matter interactions. It was shown theoretically that exposing a diatomic molecule to a laser eld can give rise to the appearance of so called light-induced CIs. In this situation the rotation is the additional nuclear degree of freedom which enables the formation of the CIs. The topology of these light-induced CIs is easily revealed by describing the molecule-light interaction in eld-dressed electronic state picture. As is well known from the eld of nonadiabatic processes, the nuclear dynamics is strongly coupled to the electronic dynamics in the vicinity of CIs, due to the appearance of innitely large nonadiabatic coupling terms. Being singular at the CI, the nonadiabatic couplings may become the source of numerous statical and dynamical phenomena. The possibility to induce articially -by using a laser eld -strong nonadiabatic eects into a molecular system promises a wealth of controllable new phenomena even in diatomic molecules.Conical intersections are abundant in free polyatomicmolecules. There is a signicant dierence between these natural CIs and the laser-induced ones. The natural CIs can not be modied while light-induced ones can be controlled. The positions of these CIs are determined by the laser frequency and the strengths of their nonadiabatic couplings by the intensity of the laser.

Figure 10 :

 10 Figure 10: Autocorrelation function and population on the adiabatic upper state surface are gured as a function of time. The applied laser eld intensity and photon energy are: I = 3.0 × 10 8 W cm 2 and ωL = 1.968 eV. For more details see Fig.

9 .

 9 In this work the topological or Berry phase in the Na 2 molecule was studied. We performed calculations along closed contours that surround a light-induced CI and obtained α = π for the value of the topological phase. As known for the case of single CIs given in nature, this result can be considered as a proof that a true conical intersection has been found. Interesting results have also been found for contours which do not surround the light-induced CI. At the heart of this work we have also performed wave packet calculations demonstrating that the light-induced CI has a strong impact on the molecular dynamics even for weak laser elds. The situation resembles that in eld free polyatomic molecules where the dynamics is highly aected by CIs given by nature owing to the strong coupling between the nuclei and the electrons.

  {(n , n)} 0,±1,±2,... . By substituting Eq.23 into Eq.21 (see the (t,t') formalism as derived by[START_REF] Peskin | [END_REF]) one gets that the solution of the TDSE is given by|Ψ(t) >= n e -iω L nt [e -iH F t/ |Ψ(t = 0) >] n,n =0 (25)and therefore when {e -iω L nt } n=0,±1,... are used as a basis where each one of the Fourier component is the eldfree initial state. For our case the (n, n ) Floquet matrix element is dened as[H F ] n ,n = δ 2n ,2n H X (R, θ)|Ψ elec

	set then
	H F ψ(t) = i	d dt	ψ(t)	(26)
				X	Ψ elec X |	(27)
	+ δ 2n +1,2n+1 H A (R, θ)|Ψ elec A	Ψ elec A |
	+ [δ 2n ,2n+1 + δ 2n+1,2n ]	E 0 2	cos(θ)d(R) •
	• [|Ψ elec A	Ψ elec X | + |Ψ elec X	Ψ elec A |]

T 0 e -iω L nt [-i ∂ ∂t + H(t)]e +iω L n t dt

where

Acknowledgment Á.V. acknowledges the OTKA Grant No. 80095. The work is partly supported by TÁMOP 4.2.1./B-09/1/KONV-2010-0007/IK/IT project. The project is implemented through the New Hungary Development Plan co-nanced by the European Social Fund, and the European Regional Development Fund. N.M. acknowledged ISF 96/07. L.S.C. acknowledges nancial support by the DFG.

Appendix

A.

On the propagation of a wave packet with a time-dependent Hamiltonian in the Floquet picture

The TDSE we wish to solve is

where H(t) = H(t + T ) stands the time-dependent Hamiltonian for a diatomic molecule in a cw laser eld with the frequency ω L = 2π/T . In our studied case