The calculation of molecular multiphoton processes using the R-matrix-Floquet method

Computer Physics Comrnunications ELSEVIER

Computer Physics Communications 114 (1998) 27-41

The calculation of molecular multiphoton processes using the R-matrix-Floquet method J. C o l g a n , D . H . G l a s s , K. H i g g i n s , P.G. B u r k e Department of Applied Mathematics and Theoretical Physics. The Queen's University of Belfast, Belfast, BT7 INN, UK

Received 10 August 1998

Abstract A unified R-matrix-Floquet theory, which previously enabled the interaction of intense laser fields with complex atoms and atomic ions to be calculated in a fully nonperturbative manner, has been extended to enable the interaction of intense laser fields with complex diatomic molecules to be treated. The theory takes advantage of the R-matrix division of configuration space into internal and external regions and treats the interaction between the laser field and the target molecule in each region using the most appropriate form of the interaction Hamiltonian. This enables standard multi-centre electron-molecule scattering programs to be modified in a straightforward way to solve the problem in the internal region and single-centre atomic multiphoton propagator methods to be extended to solve the problem in the external region. In this paper, these new theoretical and computational developments are summarised and the first multiphoton ionisation calculations for H2 are presented. (~) 1998 Elsevier Science B.V. PACS: 31.15.Ar; 33.80.Rv; 33.80.Eh Keywords: R-matrix; Multiphoton; Diatomic molecule; lonisation; Floquet-Fourier; Dissociation

1. Introduction The study of atoms and atomic ions interacting with intense laser fields has attracted considerable attention in recent years. In particular, the availability of increasingly powerful lasers has made possible the observation of many multiphoton processes. Examples of these processes include multiphoton excitation and ionisation, harmonic generation and laser-assisted electron-atom scattering. These processes have recently been reviewed by Gavrila [1], Burnett et al. [2], Mason [3] and Protopapas et al. [4]. Compared with the atomic case, the interaction of molecules with intense laser fields is considerably more complicated (see for example Bandrauk [ 5 ] ) . Firstly, the loss of spherical symmetry, due to the presence of more than one nucleus, means that the electronic structure is more difficult to treat theoretically and to interpret experimentally. Secondly, nuclear motion effects such as rotation, vibration and dissociation occur in addition to electronic processes. For example, one intense laser field effect is the modification of the nuclear potential energy curves giving rise to bond softening (or hardening) which plays an important role in multiphoton dissociation. Experimentally, the dynamics of molecules in intense laser fields have been studied extensively for many years (see for example work by Frasinski et al. [6], Codling and Frasinski [7] and Thompson et 0010-4655/98/$ - see front matter (~) 1998 Elsevier Science B.V. All fights reserved. PII S0010-4655 (98)00106-4

Colgan et al. / Computer Physics Communications 114 (1998) 27~t l

28

al. [ 8 ] ) . These experimental studies have been complemented in recent years by several detailed theoretical studies for H~- in intense laser fields (see for example Giusti-Suzor et al. [9], Plummer and McCann [ 10], Madsen and Plummer [ 11 ], and Kulander et al. [ 12]). However, very little detailed theoretical work has so far been carried out for more complex molecules. In this paper, we report the start of a programme of work to help fill this gap, by extending the R-matrix-Floquet theory and associated computer programs to enable the interaction of intense laser fields with complex diatomic molecules to be studied. The R-matrix-Floquet theory of multiphoton processes was first introduced by Burke et al. [ 13,14] to describe the interaction of a linearly polarised, monochromatic, monomode and spatially homogeneous laser field with a general atom or atomic ion. The theory is nonperturbative and is applicable to both multiphoton ionisation of atoms or ions and to laser-assisted electron-atom scattering and enables electron-electron correlation effects to be accurately treated. The theory, which also applies to the molecular systems considered in this paper, takes advantage of the fact that in the R-matrix method, configuration space is divided into two or more regions, in each of which the most appropriate form of the laser-atom or laser-molecule interaction Hamiltonian is adopted. The internal region is defined by the condition that the radial coordinates of all N + l electrons satisfy

ri<_a,

i = 1 , 2 ..... N + I ,

(1)

where we consider the interaction of a laser field with an N + 1 electron target. The sphere of radius a is chosen to just envelop the charge distribution of the N-electron residual atomic or molecular ion in multiphoton ionisation or the N-electron target atom or molecule in laser-assisted scattering. In this internal region electron exchange and electron-electron correlation effects involving all N ÷ 1 electrons are both important at low energies, and are included by adopting an antisymmetrised configuration interaction expansion of the total wave function, where the dipole length gauge is used to describe the laser-target interaction. The external region is defined by the condition that one of the N + 1 electrons lies on or outside the sphere of radius a, while the remaining 'target' electrons are confined to lie within this sphere. In this region electron exchange between this outer electron and the remaining N electrons is negligible. The dipole velocity gauge is then used to describe the laser interaction with this outer electron. Finally, at some large radius a / a transformation to the acceleration frame (Kramers [15], Henneberger [ 16]) may be carried out, enabling partial multiphoton ionisation amplitudes and scattering amplitudes in laser-assisted electron-atom or electronmolecule scattering to be defined. In order to implement the above theory for molecules we are developing a general computer program package which enables multiphoton ionisation rates for a general diatomic molecule to be calculated in situations where relativistic effects can be neglected. In the internal region this program is based on the general multi-centre UK molecular R-matrix scattering package, used for many years to describe the scattering of electrons by diatomic molecules (Burke et al. [ 17], Gillan et al. [ 18] ). In the external region, where we will see that a single-centre expansion can be adopted, the package developed by D6rr et al. [ 19] to describe multiphoton ionisation of atoms and atomic ions has been modified to treat diatomic molecules. Thus, although the computer program developments required to treat the general diatomic molecule problem are considerable, they are nevertheless feasible on a reasonable timescale by taking advantage of major program packages already developed which describe electron-molecule scattering and atomic multiphoton processes. The rest of this paper is organised as follows. In Section 2, we summarise our R-matrix-Floquet theory of molecular multiphoton processes. In Section 3, we give an overview of the computer program package that we are developing. In Section 4, we illustrate our theory and programs by presenting our first multiphoton ionisation calculations for H2, where the molecule is aligned along the laser field polarisation direction. Finally, in Section 5, we present our conclusions and consider briefly future directions of research that have been opened up by our new theory and computer programs.

J. Colgan et al. / Computer Physics Communications 114 (1998) 27-41

29

i th electron Internal Region

G= centreofgravity ofthemolecule.

External Region

I Length Gauge

Velocity Gauge

Multicentre (N+I) electrons

Acceleration

Single centre I l electron a

/

Single centre 1 electron

\

/

A G ~--- RA - -~

a

Rn--

\

B -

A

z

E

Electron-molecule co-ordinate r

Fig. 2.

Fig. 1. Fig. I. Partitioning of configuration space in the R-matrix-Floquet theory. Fig. 2. Coordinate frame of the molecule with the field polarisation along the internuclear axis.

2. R - m a t r i x theory of m o l e c u l a r multiphoton processes

2.1. Preliminaries Our R-matrix theory of molecular multiphoton processes starts from the "fixed-nuclei" approximation in which the motion of the N + 1 electrons in the target molecule is calculated in the field of the nuclei, which are assumed fixed in space. The fixed-nuclei approximation then provides the first stage in a calculation where the nuclear motion is explicitly included in a second stage. In this respect our approach is analogous to that adopted by previous R-matrix studies of electron-molecule scattering [17,18] and molecular photoionisation [20]. For each internuclear separation, configuration space is partitioned into an internal and an external region defined by Eq. (1) and illustrated in Fig. 1, where the sphere of radius a is centred on the centre-of-gravity of the molecule. In the nonrelativistic limit, the molecular system in the presence of an external laser field is described by the Schr6dinger equation

HN+I + -1Xj-'£A(ri, t) "Pi + ~ C i=1

~-'~A2(ri, t)

qt(XN+l,t) = i

qr(X~+l, t),

(2)

i=1

where we have adopted the Coulomb gauge such that the vector potential A satisfies V . A = 0. Also in Eq. (2) HN+I is the nonrelativistic part of the N + 1 electron-molecule Hamiltonian in the absence of the field and XN+I :~ ( X l , X2 . . . . . X N + I ) , where x i ~ (ri, o'i) are the space and spin coordinates of the ith electron. We assume that the laser field is monochromatic, monomode, linearly polarised and spatially homogeneous (i.e., its wavelength is large compared with the size of the molecule). Hence, we can write

A(ri, t) = A ( t ) = ~:Ao sin w t ,

(3)

where g~ is a unit vector in the polarisation direction and w is the angular frequency. The electric field £ ( t ) is then given by

£(t) -

1 d

c dt

A(t) = ~g0coswt,

(4)

where £0 = - w A o / c is the electric field strength. In accordance with current understanding of field alignment effects, we consider in this paper the situation of most experimental interest where the molecule is aligned along the laser field polarisation direction. This situation is illustrated in Fig. 2, where A and B are the nuclei, with

J. Colgan et al. / Computer Physics Communications 114 (1998) 27-41

30

charges ZA and ZB, respectively, ri is the vector position of the ith electron with respect to the centre-of-gravity G of the electron-molecule system/rod the z-axis is chosen along the laser field direction from A to B where R is the internuclear separation. 2.2. Internal region solution

As in the atomic case [ 14], we transform Eq. (2) in this region to the dipole length gauge, defined by the unitary transformation g~(Xu+l,t) =exp

--

A ( t ) • RN+t

(5)

~L(XN+l,t),

where N+I

RN+1 = Z

(6)

ri .

i=I

The wavefunction qtL(XN+,, t) then satisfies the equation [HN+, q- ~¢ (t) " RN+,] qtL (XN+I, t) = i ~ q t L (XN+,, t ) .

(7)

In order to solve this equation, we introduce the Floquet-Fourier expansion OO

qtL (XN+I, t) = e-let Z

(8)

e-in'°t-Lql, ( XN+I) •

n=--oo

Substituting this expansion into Eq. (7) and equating coefficients of e x p [ - i ( E + n w ) t ] gives (HN+,

-- E -

nw) ~p~ + ON+, (~Pn~, + ~b~+t) = 0,

(9)

where N+I 1

ON+I = ½EO~ " RN+I = ~C'OZ

Zi.

(10)

i=l

We can rewrite Eq. (9) as an infinite matrix equation in photon space as

(HF - El) q,L = 0 ,

(11)

where qt L is a vector with components .. "' ,t,L L . . . . and the Floquet Hamiltonian HF has the ~',--2, ,t,L ~'n--l, a,L ~'n, ¢,+, form

Hu+t - (n - 1)w l'I F =

DN+I 0

DN+I HN+1 -- nw DN+I

0 DN+I HN+1 -- (n + 1)w

(12)

In order to solve Eq. (9) we introduce a multi-centre basis in the internal region. Following the expansion used in electron-molecule scattering [ 17,18 ], we write

J. Colgan et al./Computer Physics Communications 114 (1998) 27-41 ~.t~n ( XN+I ) • .fit Z

"~i (X1 ..... XN, O'u~- 1, R ) "Oj ( r N + l ) aijkn + ~ ij

Xi (Xl ..... XU+l; R ) bikn ,

31 (13)

i

where A is the antisymmetrisation operator, the ~i are formed by coupling the electronic wavefunction ~i of the residual molecular ion for fixed internuclear separation R with the spin function of the ejected electron to form an eigenfunction of S 2 and Sz. the rlj are continuum functions describing the motion of the ejected electron, whose form is described in [ 17]. and Xi are N + 1 electron bound configurations constructed from the bound molecular orbitals which vanish by the boundary of the internal region. Finally. the coefficients auk. and bikn in Eq. (13) are obtained by diagonalising the operator HF q- Lb in the basis 0 L, k whose components are I# L k. , giving

( q,Lk ] H v + L b l O f ' ) i n t = E k 6 k k

',

(14)

where the integral is taken over all electron coordinates in the internal region. Also in Eq. (14) the Bloch operator Lb [21] is diagonal in photon space and is defined by the diagonal components 1 Lb=~

/ 3 ~r i

~(ri--a)

b- 1 ri

15)

'

i=1

where b is an arbitrary constant. This operator ensures that HF + Lb is Hermitian in the internal region in the space of functions satisfying arbitrary boundary conditions at r = a, and hence the energy Ek in Eq. (14) is real. We now rewrite Eq. (11 ) in the form (16)

( H F q'- Lb -- EI)~O L = L b O L •

Using the spectral representation of the operator (HF + Lb -- EI) given by Eq. (14), we then obtain

I~L)=~-~.I

OL\

1

/0z,

k/E---~-2_E~ k tL01e'L) •

(17)

k

We project this equation onto the channel functions defined by [ ~ i × }~,m,,] ~ [ ~ i ( X l ..... XN, O'N+I;R) X Y~m,ti(l'N+l)] ,

(18)

where ]~,,,~, is a spherical harmonic, and onto the nth component in photon space. Evaluating on the boundary rN+l = a then yields

Rigime,ni,~frn#n,(E)

Fi,gfm,in,

i'g~m':in'

bFi, em, .,~

(19)

, e, f ru+l=a

where we have introduced the R-matrix in the dipole length gauge. wL . L igimeink wi'g: m~in' k

l p

(20)

k

and the reduced radial wavefunctions Fie~m~,.(r) L by -1 L , rN+lFigimen (rN+l)

=

( [ ~ i X Ygimei] ] ~ 0 .L) t

.

(21)

J. Colgan et al. / Computer Physics Communications 114 (1998) 27-41

32

The prime in this equation means that the integral is taken over all electronic coordinates in the internal region except the radial coordinate of the (N + 1)th electron. Finally, the surface amplitudes wiei,,~ , L nk in Eq. (20) are defined by a - l w Lie+,,c?k = ([~i X Ye,me,] [ ~0/',)'ru+~=a•

(22)

Eqs. (19) and (20) are the basic equations describing the solution of Eq. (2) in the internal region and provide the boundary conditions for the external region solution. We note that since we have chosen the molecular axis to be aligned along the field polarisation direction, as in Fig. 2, then the total z component of the orbital angular momentum of the molecular system A = Ai ÷ me, is conserved so that

Ai + me, = a I + rn~, ,

(23)

where hi is the z component of the orbital angular momentum of the residual molecular ion and me, is the z component of the orbital angular momentum of the ejected electron.

2.3. External region solution In this region we transform Eq. (2) to the dipole velocity gauge for the ejected electron, while treating the remaining N residual electrons still in the dipole length gauge. The corresponding unitarity transformation is given by g~ (Xo+l,t) = exp ( - i A (t) . RN - i f A 2 (t') dt' ) ~ v ( X o + , , t ) c ~

'

(24)

where O

R N -~ ~

ri ,

(25)

i=1

The wavefunction g~v (Xo+l,t) then satisfies the equation

(H°+I + E (t) " RN + I A ( t) "P°+') ~v (Xu+l't) = i O ~v (X°+l't)

(26)

As in the internal region, we introduce the Floquet-Fourier expansion OO

qtV(xN+l't) =e-iEvt Z

e--in°~t~tVn ( X N + I ) .

?l = --

(27)

OO

Substituting this equation into Eq. (26) and equating the coefficients of exp [ - i (Ev + no)) t] gives (HN+I -

EV -- no+) ~pv + Do (~l,Vn_, +

0 . +v , )

+ i ~Ao c

e.

Po+, (g'Vl

- ~nv+l) = 0 ,

(28)

where N

DN = ½gO~" RN

1

~gO Z

Zi.

i=1

Again, as in the internal region, we can rewrite Eq. (28) as an infinite matrix equation in photon space.

(29)

J. Colgan et al. / Computer Physics Communications 114 (1998) 27-41

33

In order to solve Eq. (28), we introduce the following close coupling expansion for the components ~pv. ?/

~-IV ( X N + I ) = Z " ~ i ( X igime,

1. . . . . XN; O'N+ 1 , R ) rN+lGigimei -1 v n ( r N + l ) Yg,mei ( r N + l ) ,

•

(30)

where the summation over i goes over the same range as in Eq. (13), but where now the antisymmetrisation operator ,A is omitted since electron exchange effects between the ejected electron and the residual molecule vanish because they occupy different regions of space and where the N + 1 electron bound configurations are zero for the same reason. Substituting Eq. (30) into Eq. (28) and projecting onto the channel functions defined by Eq (18) and onto the nth component of the wave function in photon space then yields the following set of coupled differential equations: d2 -d-rr2

gi (gi + 1) 2 (Z - N) ~n) Gv r2 + + k igimei n ( r ) = 2 Z r •i g t t t imgin

Wig, n' (r)Gi,g;m,¢, n, ( r ) v mcni'gfm~ i t

~

r > a. --

t

(311 In this equation Z = ZA + ZB, (32)

k~n = 2 ( Ev - wi + noJ ) ,

where wi is the energy of the ith residual molecular ion state defined by

<~i IHNI ~j> = wi,~i: ,

(33)

and wiV~,m~jni,g;,%,, ( r ) is the long-range potential, coupling the channels. As in the atomic case [ 14], the long-range potential is the sum of three terms, (34)

WiV mei hi'g:mr,n,(r) = We + Wo + We

where We arises from the electronic Hamiltonian term HN+I in Eq (28), Wo arises from the dipole operator term DN in Eq. (28) and We arises from the momentum operator term ~ • PN+I in Eq. (28). We give explicit expressions for We, 'No and We in Appendix A. 2.4. Matching the internal and external regions solutions

In order to solve Eq. (31 ) in the external region r > a, we need to determine the boundary condition satisfied v n (r) at r = a. This boundary condition can be obtained from the R-matrix, defined by by the functions Gigging, Eq. (20), which relates the function Fi~,m,,n ( r ) to its derivative on the boundary by Eq. (19) and from the relationship between E i£imti L n ( r ) and Gig, v me,,: ( r ). The relationship between F/g.me, t~ . (r) and Gig, v m~i. ( r ) can be obtained from Eqs. (5) and (24), which gives

~'v (XN+1, t) = exp

(J i2 ~-5c

A 2 ( t l ) d t ' - ic A (t) • rN+l

>

~L (Xu+t, t)

(35)

where ri < a,

i = 1 ..... N; rN+l = a.

Substituting for A (t) from Eq. (3) gives

(36)

34

J. Colgan et al./Computer Physics Communications 114 (1998) 27-41 g"v(Xu+l,t)=exp

(A2)

(

A2

)

i~cOzt exp --i 8o~c "o 2 sin (2~ot) - iAo~ - - "rN+l sino)t g~c (XN+I,t) . C

(37)

Using the Floquet-Fourier expansions (8) and (27) then enables Eq. (37) to be written as OO

OO

e -iEvt ~_~ e - i m o t ~ V ( X N + l ) : e - i ( E - E e ) t

Z

n=-- O0

<9O

fg(Ao,~.rN+l)e-ig°Jt

~ = - - Oco

Z

e--imotlpL(XN+l) '

(38)

tl=-- O0

where Ep = A~/4c 2 is the ponderomotive energy associated with the ejected electron, and

fe(A°'£:'ru+l)=

o+ ( A 2 x~ (_~^ ) Z Jr' k,8wcZjJe-2t' ~''rU+l ,

(39)

e~=--cx:,

where we have used the well-known result

e-izsinO = ~

(_l)njn(Z)e

inO.

(40)

n = -- 00

From Eq. (38) we then obtain Ev = E - Ep and

~9vn (XN+I) = ~

fn-n' (Ao,~'r~+l) ~

(XN+I).

(41)

nt=--~

The relationship between Fie,,,ein L v n (a) can be obtained by projecting Eq. (41) onto the channel (a) and Gigimei functions defined by Eq. (18), using Eqs. (21) and (30), and evaluating on the boundary r = a. We obtain

aV iglmeln (a) :

L Z Affimeini'g;m~in' Fi'~;m~in' (a) , i'g;m~i n'

(42)

where we can show that the transformation matrix

Aieme,ni'+;m n' =

Z J+ 8o,c2]

~

~'=--oo +1

×J -t

gi

(X) P'g; (X) Jn-n'-2g

ax

dx(~ss,¢~MsMrsf~SiSfC~MsiM~i~rnirn;¢~Ai~;(~meim~,¢~ii ,,

(43)

where Si and Ms, are the spin quantum numbers of the residual molecular ion states. We see that the transformation matrix is diagonal in the quantum numbers of these states. Using Eq. (42) and the R-matrix equation (19), obtained from the internal regional solution, the relation between G l¢ittlgi v I1 and dG/~i m ¢ i n /dr on the boundary r = a can be obtained as described by Burke et al. [ 14]. Eq. (31 ) can then be integrated outwards from r = a to a radius r = g where an asymptotic expansion can be applied as described in Section 3.2.

J. Colgan et al./Computer Physics Communications 114 (1998) 27-41

35

3. Overview of the computer program package 3.1. I n t e r n a l region

We consider in this section the situation described by Eq. (23) where the molecule is aligned along the laser field polarisation direction so that the projection of orbital angular momentum A along this axis is conserved• The Floquet Hamiltonian H F matrix then has the form

1t F =

[HN+l]uu -- (tl -- 1)w

[DN+I]ug

0

[DN+I ]gu

[HN+I ]gg -- llW [ D N + I lug

[DN+I ]gu

0

(44)

[HN+I ]uu -- ( n + 1)w

where we have used Eqs. (12), (13) and (14). In Eq. (44), the Hamiltonian matrix elements are given by

[HN+l]gg

= (l~g+lnN+lll~g) ,

[HN+l]uu

+ I ~ N + l [ ~1 . ) +, = ( 1~u

(45)

and the dipole matrix elements are given by [DN+I] tug ----( l~,g ~ u ) 1, +

(46)

where we consider here the situation where the (N + 1)-electron molecular target is initially in the 15+ state, although the program has been written for arbitrary initial state. The dagger in Eq. (46) refers to the Hermitian conjugate. No program development was necessary in order to evaluate the Hamiltonian matrices as these can be calculated using the standard electron-molecule scattering package [18]. Consequently, they will not be discussed any further here. However, we had to develop a new program to determine the off-diagonal dipole matrices. The theory underlying this new 'dipole' program is described below. In order to understand the operation of the dipole program, we consider the Floquet Hamiltonian given above: we are required to calculate dipole matrix elements of the form

/ I I t I ( X I ..... X N + l ) ( ~ 'i=1 'ri)

.....

XN+l)dXl'"dXN+l'

(47)

where I and J are the i ~g+ and 1 ~u+ scattering states• The wavefunctions 1/)'1 and ~s are each expanded as a linear combination of products of (N + 1) orthogonal multi-centre molecular spin orbitals ~b,, such that

~It' = Z

ai(f~li~)2i'"~(N+l)i) ,

ii~,J = ~

i

t t t bi(~Oli(l~2i...(~(N+l)i)

,

(48)

i

where the coefficients ai and bi are such that the wavefunctions are normalised and antisymmetrised with respect to the interchange of any two electrons. The integral given by Eq. (47) then becomes

~"~Zaebjf(q~lpq~2p...dp(N+l)p)(~.ri)(C~tljq~t2j...qb~N+l)j)dxl...dXN+ p

j

1.

(49)

\ i=l

The dipole operator is a one-electron operator and hence when there is a difference of more than one spin orbital between the two wavefunctions, the orthogonality of the spin orbitals implies that the integral defined

J. Colgan et al. / Computer Physics Communications 114 (1998) 27-41

36

by Eq. (49) is identically zero. If we examine the case where the wavefunctions differ by one spin orbital, which we shall denote as ~bk where 1 < k < N ÷ 1, then the integral reduces to

Z ~ apbjPpj f qbkp ( ~ - rk) ~'kjdxk , where

p

J

ppj

is a first-order density matrix defined as

ppj = f (~lp...¢b<~-l)pCb¢k+l)p...fb¢N+l)p) (4~lj.-.&¢k--1)j&¢k+~)j.-.q~¢N+l)j)

(50)

dXj...X~_IXk+I...XN+I.

(51)

In any calculation, all nonzero elements of the dipole matrix arise from wavefunctions which differ by one spin orbital. Hence, we were required to develop a program which evaluates the density matrix ppj for a pair of scattering wavefunctions and then sums over the product of this density matrix with the corresponding integral in accordance with Eq. (50). The final version of the program which completes this task is referred to as DENPROP. Firstly, DENPROP assigns a unique number to each spin orbital. Each scattering wavefunction is then expressed as sets of numbers, each set corresponding to the assignment of (N ÷ 1) electrons to specific spin orbitals. DENPROP then compares each set of numbers in the first scattering wavefunction with each set of numbers in the second scattering wavefunction. If the sets differ from one another by two or more, the corresponding dipole matrix element is set to zero in accordance with the arguments given above. However, if they differ by only one, the calculation proceeds in two stages: firstly the density matrix given by Eq. (51 ) is determined and secondly the integral in Eq. (50) which involves the two differing orbitals is evaluated. The second step required very little program modification since a set of routines already existed in the standard electron-molecule scattering program to calculate molecular integrals of the form

J dp(r" (cos0) i (sin0) j Y/m)~b' d x ,

(52)

where the (r n (cos0) i (sin0)J Ylm) are one-electron property operators in which n,i and j are integers and the Yr,, are the usual spherical harmonics. By selecting the appropriate values for the integers n, i and j and including the electric field strength Co we were able to calculate integrals of the form we required.

3.2. External region The solution in the external region proceeds by adopting the same approach as that of D6rr et al. [ 19], who described the solution in this region in the velocity gauge for the atomic multiphoton case. This is appropriate for the present work since a single-centre expansion has been used in the external region and it greatly simplifies the programming since the same computer programs can be used with only small modifications. A brief outline of this work is now given. The system of coupled differential equations which define the problem in the external region as given in Eq. (31) can be written in matrix form as

+

dr + V ( r ) + k 2 G(r)=O,

(53)

where the derivative term Pd/dr is due to the interaction of the field with the ejected electron and arises from the We term in Eq. (34) and V contains all the nonderivative potentials and can be written as a multipole expansion. The component of the potential WD in Eq. (34), which arises from the interaction of the field with the target electrons, and is independent of r, is diagonalised and included in k 2 so that V only contains off-diagonal couplings which vanish as r --+ c¢. This diagonalisation corresponds to a 'dressing' of the target by the field.

J. Colgan et al. / Computer Physics Communications 114 (1998) 27-41

37

The first part of the solution in the external region involves transforming the R-matrix from the length to the velocity gauge at the boundary r = a as described in Section 2.4 and then propagating it to some further distance r = a t. This propagation involves carrying out a transformation which removes the first derivative term from Eq. (53) so that the standard Light-Walker propagator [22] can be used. The details of this transformation and the resulting modified potential is given by D6rr et al. [ 19]. At r - a ~, Eq. (53) is solved by using a version of the asymptotic expansion developed by Burke and Schey [23] which has been modified to take account of the derivative and multipole expansion terms. This solution is subject to outgoing wave boundary conditions at r ---, ~x~ which have been formulated in the acceleration frame since in this frame the channels are asymptotically uncoupled. Finally, the solutions obtained are matched at r = a' with those obtained by propagation and then total ionisation rates, branching ratios and angular distributions can be calculated.

4. Results In this section we present the first molecular multiphoton results for H2 using our new theory and computer programs. In order to represent the H~- residual ion, a two-state approximation was adopted corresponding to the X 2Eg+ ground state and the first 2~+ excited state. These states were represented by LCAO-MO-SCF wavefunctions constructed from a ls-2s-2p STO ~r basis as used by Tennyson et al. [24] in the context of electron-scattering from H + at low energies. Most of the results presented here were carried out within the two-state close-coupling approximation, where the residual electron is confined to the lo-g or lo-u orbital. For each of the states retained in the calculation ( I Eg, + IE~-) all partial waves up to and including g : 5 were included. Increasing this to g = 7 changed the total ionisation rate by no more than five percent for the highest intensity considered (1013 W cm -2) indicating convergence for g = 5. In addition, five Floquet blocks, including three for absorption and one for emission, were found to be sufficient for convergence. In all the two-state results, a fixed internuclear separation of 2 a.u. and an R-matrix internal region radius of 10 a.u. have been used, while propagation to 30 a.u. in the external region was found to give convergence in most cases. The two-state approximation gave an energy for the H2 ground state of - 0 . 5 1 9 8 2 a.u. relative to the ground state of H~-. The potential curves for the system are illustrated in Fig. 3. Having described the approximation used for the calculations, a preliminary check was carried out using a one-state approximation (the x zE~ ground state). The total ionisation rate from the ground state of H2 was calculated at an intensity of 10 l° W cm -2 for several frequencies where one photon is sufficient to ionise the system. This was done for values of the internuclear separation increasing from R = 1 a.u. in steps of 0.2 a.u. and, for each photoelectron energy, the results compared with multiphoton calculations for the ionisation of He using a one state ( l s ) approximation. These results are presented in Fig. 4. The convergence of the results and the consistency with the He calculations gives us confidence to proceed to the two-state calculations. Fig. 5 shows results for ionisation of H2 at an intensity of 1011 W cm -2 using the two-state approximation described above for R = 2 a.u. A frequency range is considered where the ionisation rate is influenced by single-photon excitation of doubly-excited states. It can be seen that there is a Rydberg series of 1Eu+ resonance states converging to the 2E~- threshold, although it must be noted that our results are not accurate close to threshold since greater propagation distances are required. The importance of such states in the photoionisation process has been studied by Tennyson et al. [20] who used a straightforward single photon R-matrix approach. The positions and widths of the lowest two states are in reasonable agreement with those obtained from the scattering calculations of Tennyson et al. [24] although the nuclear motion would need to be taken into account for a more detailed comparison. Fig. 6 considers calculations at an intensity of l0 II W cm -2 for higher frequencies where absorption of a single photon can leave the residual molecular ion in the 2Eu+ excited state as well as the 2Eg+ ground state. Since this state is repulsive this corresponds to photodissociation of the molecule. It is found that this process

J. Colgan et al. /Computer Physics Communications 114 (1998) 27-41

38

It

'x~,H zl rgl

.~ g

;

x+

z

< - - ~

~ - - - -.-.:.:. .2. - - -

l

l

[

~

l

2 S -

h

,

lH

RI%) Fig. 3. Potential curves for H2 and H + + The dashed lines represent R y d b e r g b o u n d states converging to HE ( X 2E+ ) and R y d b e r g resonance states converging to H 2+(2Eu+).

iI

. . . . .

i

i 1.0 i

\

1

0,8

3eV

,~ 0.6

Io

4eV ~

0.4

5eV

0.2 + \ .

1 7eV ]

I

\ • 0.0 [ .

0.0

.

Helium Limit as R - > 0 . . . . . .

!

. . . 1.0 2.0 [nternuclear Separation, R (a.u.) Fig. 4.

.

3.0

0.80

0.85

0.90 Frequency (a.u.)

0.95

Fig. 5.

Fig. 4. The total ionisation rate against internuclear separation R at an intensity o f 10 t° W c m - 2 for fixed values o f the photoelectron energy as shown. The values for R > 1 a.u. were obtained using the present a p p r o a c h while those at R = 0 a.u. were obtained f r o m a He calculation u s i n g the atomic code. Fig. 5. The ionisation rate against laser frequency for R = 2 a.u. at 10 jl W c m - 2 in the range where a single photon excites 1~+ d o u b l y excited states.

makes a smaller contribution to the total ionisation rate than that leaving the residual molecular ion in its ground state. In considering multiphoton ionisation, resonances have also been found due to one-photon resonant twophoton ionisation leaving the residual ion in its ground state. A Rydberg series of such resonances can be found converging to the 2Xg+ threshold and more details of these will be presented in future work. Here we consider just one such resonance. The laser frequency corresponding to one-photon excitation of the l 2£u+ bound state is almost the same as that necessary for two-photon excitation of the second l Xg+ doubly excited state. The

J. Colgan et al. / Computer Physics Communications 114 (1998) 27-41

39

0.0050 . . . . . . .

4F

0.0040 ~3

0,0030

~2 g

~2

g

0.0020

"2

Ol

0,0010

1,0

1.1

1.2 Frequency (a.u.) Fig. 6.

1.3

1.4

0.0000

-0.530

-0,525

-0.52

2 1 _ -0,515

-0.510

_

l _ _ -0.505 -0.500

Real(E) (a.u.) Fig. 7.

Fig. 6. This plot is a continuation of Fig. 5 for higher laser frequencies where now absorption of one photon is sufficient to bring about dissociative ionisation. Total ionisation rate (solid line), partial rate leaving residual ion in its ground state (dashed line) and partial rate for dissociative ionisation (dotted line). Fig. 7. One- and two-photon LIDS for R = 2 a.u. The arrows and crosses indicated the bound states and doubly excited state respectively at low intensity. Trajectory of states at w = 0.4545 a.u. (solid line), at o) = 0.4546 a.u. (dashed line), at o) = 0.45195 a.u. (dotted line) and at o) = 0.45198 a.u. (chain line). For each trajectory the intensity is increased from 0 to 1013 W cm -2.

investigation of such a double resonance has been studied for one- and three-photon processes in He by Glass et al. [25] where substantial enhancement of the total ionisation rate was found. Many comparisons could be made between the work on He and the present results, but here we focus on the possibility of laser-induced degenerate states ( L I D S ) . LIDS occur when two atomic (or molecular) states dressed by the laser field become degenerate for particular values of the frequency and intensity. Such states have been studied in detail using the R-matrix Floquet approach in H - [26], Ar [26,27] and Mg [28] and have been found to be a common feature of atoms in intense laser fields. The calculation of two LIDS processes in the two-photon ionisation of H2 is shown in Fig. 7. Since there is such a strong mixing of all three states (the 1Eg+ ground state, the 1Eu+ bound state and t h e l ~ doubly excited state) it is difficult to identify the states except at low intensity. Nevertheless, the behaviour associated with LIDS is clearly present. At a frequency of 0.4545 a.u. {he ionisation rate from the state, which at low intensity is the doubly excited state, decreases as the intensity increases. For the same frequency the rate from the state, which at low intensity is the ground state, increases initially but begins to level off at a width of just above 0.002 a.u. At a frequency of 0.4546 a.u. the states can be seen to switch over with the ground state at high intensity taking on the behaviour previously associated with the doubly excited state and vice-versa. This degeneracy occurs between these two frequencies and for an intensity of 9 x 1011 W cm -2. A similar process occurs at a slightly lower frequency (0.45195-0.45198 a.u.) and at the lower intensity of 8 x 10 I° W cm -2. In this case the states involved can be described at low intensity as the 1E~- singly excited state and the 11~+ doubly excited state, but otherwise the behaviour is much the same. One difference is that in this case the behaviour of the doubly excited state at the lower frequency is different from that discussed earlier. After its width decreases initially as a function of intensity it begins to increase again. This may be due to a stronger mixing of the three states.

J. Colgan et al. / Computer Physics Communications 114 (1998) 27-41

40

5. Conclusions and future directions of research We have described in this paper the start of a major new programme of research to enable the interaction of intense laser fields with complex diatomic molecules to be calculated. The first results for the H2 molecule, presented in this paper, illustrate the R-matrix theory and the computer program package that we have developed. In the future, we intend to extend this research in a number of directions. Firstly, we plan to incorporate into our computer program the extension to enable the molecule to be oriented in an arbitrary direction relative to the laser field polarisation direction. This will enable more complete comparisons to be made with single photoionisation experiments and theory and enable us to explore the importance of molecular non-alignment effects in intense laser fields. The next major step is to incorporate the nuclear motion effects, including dissociation, into our computer program. In this work we will be able to take advantage of the developments made over many years in incorporating nuclear motion effects in the R-matrix theory and programs for electron-molecule scattering. Also we plan to look at multiphoton processes for other diatomic molecules of current experimental interest such as N2 and CO. Finally, we are encouraged by recent developments reported elsewhere in this volume [ 29 ], where the UK R-matrix electron-molecule programs have been extended to treat simple polyatomics such as H20, CO2 and C2H4. We believe that these developments make the calculation of multiphoton processes involving polyatomic molecules a practical reality in the not-too-distant future.

Acknowledgements We acknowledge with thanks the support from a rolling grant awarded by the EPSRC to the Department of Applied Mathematics and Theoretical Physics at Queen's University. One of us (JC) wishes to acknowledge with thanks the award of a research studentship by the Department of Education in Northern Ireland. Finally, we wish to thank the UK electron-molecule scattering collaboration, which developed the UK molecular R-matrix scattering package over the last 20 years, supported by CCP2, and the EC atomic multiphoton collaboration, which developed the multiphoton asymptotic package over the last 8 years, supported by the EC-Science and HCM programmes.

Appendix A The evaluation of the long-range potential terms, We, WD and Wp in Eq. (34), is straightforward and we only summarise the results here. We note that in this work we use the Condon and Shortley phase convention for the spherical harmonics, in contrast to the earlier work on atomic multiphoton processes [14] where the Fano-Racah phase convention was adopted. The WE term can be written as OO

A i'l~m~ r--A--16AA,6nn, ailim

WE =

J

(A.1)

i

,~=1

where _a

, ,

ailmztfl'l~mt,=

[2li-k- l ] 1/2 (Xl ..... XN; R ) ) [_21~+ lJ ( l i m t ,h m l l l m ~, )(liOaO[liO)(rul+t~Pi(Xl ..... x N ; R ) IUm[rN+t~i, A --1 X (~SS' (~MsMs, (~SiS[~Msi Ms; •mim; ,

with

( A.2 )

J. Colgan et al./Computer Physics Communications 114 (1998) 27-41 = I

4~r

]1/2 ZNr J ~ Y 2 1 m ( I ' J ) - Z A ( - R A ) j=l

41

A~mO - Z B ( R B ) A ~ m O

(A.3)

T h e q u a n t i t i e s ( a b c d le f) in Eq. ( A . 2 ) are C l e b s c h - G o r d a n c o e f f i c i e n t s w h i c h e n s u r e t h a t m --- m !t, - rnt,. T h e WD t e r m c a n b e w r i t t e n as

WD = (qbilDN IcDi' ) ( (~nn' - 1 + ~nn' + 1 ) ~SS' ~Ms M~sf~SiS; ~Ms M~ ~m m' ~l I( 8n .m~ ~A A! ,

( A.4 )

where DN is the target dipole operatordefinedby Eq. (29). The Wp term can be written as

we=A°(~n"'+l--6"n'-l)2c l,

[(21;+1)(2l;+3)1'/2 ~r(a l,+l) ]

-+ [(2/~ - 1)(21~ + 1 ) ] 1/2

~T +

r

6t, t;-I

6t, l;+~

6SS'~MsM~.,6S~S!~gs~M~ , • 6.,,m;~m,,~6a,~;6y,

.

(A.5) where "Yi specifies the remaining quantum numbers required to define the residual molecular ion state.

References [ 1] [21 [31 141 [5] [6] [71 [81 [91 [101 [1l] [12] [13] I14] [15] [16] [17] [18] [19] [20] [21] 122] [23] 1241 [251 [261 [27] [28] [29]

M. Gavrila, ed., Atoms in Intense Laser Fields (Academic Press, New York, 1992). K. Burnett, V.C. Reed, EL. Knight, J. Phys. B 26 (1993) 561. N.J. Mason, Rep. Prog. Phys. 56 (1993) 1275. M. Protopapas, C.H. Keitel, EL. Knight, Rep. Prog. Phys. 60 (1997) 389. A.D. Bandrauk, ed., Molecules in Laser Fields (Dekker, New York, 1994) L.J. Frasinski, K. Codling, P. Hatherly, J. Barr, I.N. Ross, W.T. Toner, Phys. Rev. Lett. 58 (1987) 2424. K. Codling, L.J. Fmsinski, J. Phys. B 26 (1993) 783. M.R. Thompson, M.K. Thomas, P.F. Taday, J,H. Posthumus, A.J. Langley, L.J. Frasinski, K. Codling, J. Phys. B 30 (1997) 5755. A. Giusti-Suzor, F.H. Mies, L.E Di Mauro, E. Charron, B. Yang, J. Phys. B 28 (1995) 309. M. Plummer, J.F. McCann, J. Phys. B 28 (1995) 4073. L.B. Madsen, M. Plummer, J. Phys. B 31 (1988) 87. K.C. Kulander, F.H. Mies, K.L. Schafer, Phys. Rev. A 53 (1996) 2562. EG. Burke, P. Francken, C.J. Joachain, Europhys. Len. 3 (1990) 617. P.G. Burke, P. Francken, C.J. Joachain, J. Phys. B 24 (1991) 761. H.A. Kramers, Collected Scientific Papers (North-HoUand, Amsterdam, 1956) p. 272. W.C. Henneberger, Phys. Rev. Lett. 2l (1968) 838. P.G. Burke, I. Mackey, I. Shimamura, J. Phys. B 10 (1977) 2497. C.J. Gillan, J. Tennyson, P.G. Burke, in: Computational Methods for Electron-Molecule Collisions, W.M. Huo, F.A. Gianturco, eds. (Plenum Press, New York, London, 1995) p. 239. M. D6rr, M. Terao-Dunseath, J. Purvis, C.J. Noble, P.G. Burke, C.J. Joachain, J. Phys. B 25 (1992) 2809. J. Tennyson, C.J. Noble, P.G. Burke, Int. J. Quantum Chem. 29 (1986) 1033. C. Bloch, Nucl. Phys. 4 (1957) 503. J.C. Light, R.B. Walker, J. Chem. Phys. 65, 4272. P.G. Burke, H.M. Schey, Phys. Rev. 126 (1962) 147. J. Tennyson, C.J. Noble, S. Salvini, J. Phys. B 17 (1984) 905. D.H. Glass, P,G. Burke, H.W. van der Hart, C.J. Noble, J. Phys. B 30 (1997) 3801. O. Latinne, N.J. Kylstra, M. D6rr, J. Purvis, M. Terao-Dunseath, C.J. Joachain, P.G. Burke, C.J. Noble, Phys. Rev. Lett. 74 (1995) 46. A. Cyr, O. Latinne, P.G. Burke J. Phys. B 30 (1997) 659. N.J. Kylstra, H.W. van der Hart, P.G. Burke, C.J. Joachain, J. Phys. B 31 (1998) 3089. L.A. Morgan, J. Tennyson, C.J. Gillan, Comput. Phys. Commun. 114 (1998) 120, this issue.