# Effects of electronic correlation on the high harmonic generation in helium:
a time-dependent configuration interaction singles vs time-dependent full
configuration interaction study

Aleksander P. Woźniak<EMAIL_ADDRESS>Faculty of Chemistry,
University of Warsaw, Pasteura 1, 02-093 Warsaw, Poland  Michał Przybytek
Faculty of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warsaw, Poland
Maciej Lewenstein  ICFO - Institut de Ciencies Fotoniques, The Barcelona
Institute of Science and Technology, Av. Carl Friedrich Gauss 3, 08860
Castelldefels (Barcelona), Spain  ICREA, Pg. Lluís Companys 23, 08010
Barcelona, Spain  Robert Moszyński  Faculty of Chemistry, University of
Warsaw, Pasteura 1, 02-093 Warsaw, Poland

###### Abstract

In this paper, we investigate the effects of full electronic correlation on
the high harmonic generation in the helium atom subjected to laser pulses of
extremely high intensity. To do this, we perform real-time propagations of the
helium atom wavefunction using quantum chemistry methods coupled to Gaussian
basis sets. The calculations are done within the real-time time-dependent
configuration interaction framework, at two levels of theory: time-dependent
configuration interation with single excitations (TD-CIS, uncorrelated method)
and time-dependent full configuration interaction (TD-FCI, fully correlated
method), and analyse obtained HHG spectra. The electronic wavefunction is
expanded in Dunning basis sets supplemented with functions adapted to
describing highly excited continuum states. We also compare the TD-CI results
with grid-based propagations of the helium atom within the single-active-
electron approximation. Our results show when including the dynamical electron
correlation, a noticeable improvement to the description of HHG can be
achieved, in terms of e.g. a more constant intensity in the lower energy part
of the harmonic plateau. However, such effects can be captured only if the
basis set used suffices to reproduce the most basic features, such as the HHG
cutoff position, at the uncorrelated level of theory.

††preprint: APS/123-QED

##  I Introduction

Attoscience is a rapidly developing area of physics that studies light-matter
interactions of atoms and molecules subjected to intense laser fields. One of
the key aspects of attoscience is the high harmonic generation (HHG), which
provides ultrashort pulses of coherent radiation of wavelength ranging from
visible light to soft X-ray [1, 2, 3]. Such pulses have recently became an
extremely powerful tool for studying electronic dynamics in atoms and
molecules[4, 5, 6, 7], charge transfer reactions [8], molecular imaging [9,
10, 11] and many more. According to the famous three-step model [12, 13, 14],
during the HHG process the electron is (i) removed from the atom or molecule
by the strong laser field via tunneling ionization, (ii) accelerated away in
the field until the field sign is changed and then reaccelerated towards the
residual ion, and (iii) recombined with the ion, which is accompanied by the
production of a high energy photon. The HHG spectrum is characterized by a
long plateau region, composed of consecutive harmonic peaks, which ends with
an abrupt cutoff at the energy $E_{\textrm{cut}}$ connected to maximum kinetic
energy an electron can gain in the electromagnetic field.

HHG has been theoretically studied for decades using a variety of methods [15,
16, 17, 18, 19]. Over the last few years, as both the theoretical and
experimental attoscience shifts towards increasingly more complex systems
including solids [20, 21] and biomolecules [22, 23, 24], the real-time
propagation of the wavefunction using time-dependent quantum chemical methods
coupled to $L^{2}$-integrable basis sets have gained some recognition in
describing the electron dynamics during attosecond processes [25, 26, 27, 28,
19]. Due to a better scaling with the number of electrons and much easier
handling of multicenter systems, they are slowly surpassing the purely
numerical, grid-based approaches.

The approaches known from quantum chemistry that have been extended to the
time-domain can be generally classified into two categories: the orbital-based
ones, which include time-dependent Hartree-Fock (TD-HF) [29, 30, 31, 32, 33,
34, 35] and time-dependent density functional theory (TD-DFT) [36, 37, 38, 39,
35, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57,
58, 28, 59], and the wavefunction-based ones, such as time-dependent
configuration interaction (TD-CI) [60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70,
71, 44, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 27, 28,
87, 59], time-dependent multiconfigurational self consistent field (TD-MCSCF)
[88, 89, 90, 91, 92, 74, 93, 94, 95] and time-dependent coupled cluster (TD-
CC) [96, 97, 98, 99, 100, 101, 102, 103]. Out of all these methods the one
that gained an exceptional popularity in theoretical attoscience is the time-
dependent configuration interaction with single excitations (TD-CIS) [60, 61,
63, 64, 65, 67, 68, 69, 70, 44, 75, 76, 78, 79, 82, 83, 84, 85, 86, 27, 28,
87, 59], due to its simple formalism and low computational costs. In TD-CIS
the time-dependent wavefunction is expanded in (time-independent) CIS
eigenstates obtained via diagonalization of an atomic or molecular Hamiltonian
matrix that consists of matrix elements between the Hartree-Fock ground state
configuration and singly excited configurations (the TD-CI formalism will be
detailed more comprehensively in the next section). TD-CIS has proven useful
in predicting the HHG spectra in molecules as large as chlorinated
hydrocarbons [83] and nucleobases [87]. However, since the CIS method is an
equivalent of the Hartree-Fock method for excited states, it completely lacks
the electronic correlation effects, that are considered by many to play an
important role in laser-driven electron dynamics [104, 105, 106]. A natural
way to account for these effects would be to include higher excitations in the
CI wavefunction, starting with CI with single and double excitations (CISD).
The TD-CISD approach has been discussed several times in the literature [62,
107] and tested for two-electron systems in small basis sets [64, 65, 27], but
is hardly applied in the strong field calculations due for reasons. First,
inclusion of doubly excited configurations in the CI Hamiltonian matrix
drastically increases its size, making both the obtainment of the CI
eigenstates and the real-time propagation far more expensive and time
consuming, if feasible at all. Secondly, many authors base their real-time
propagation codes on results from existing electronic structure packages, such
as Gaussian [75, 76], GAMESS [85] or Q-Chem [44, 79, 84]. Most of these
programs usually perform the diagonalization of large matrices using iterative
algorithms, that provide no more than the ground and a few excited electronic
states, while the real-time propagation usually requires a full Hamiltonian
eigenspectrum.

Nevertheless, there have been some other attempts to account for the
electronic correlation effects in HHG. A notable example is the development of
the TD-CIS(D) method, which treats the effects of double excitations via
perturbational corrections to CIS energies [62, 64, 65, 66, 70, 44].
Unfortunately, as reported by Sonk et al. [70], it may give an erratic optical
response in intense fields. Probably the most common way relies on using TD-
DFT methods with an exchange-correlation potential of choice [39, 35, 41, 42,
43, 44, 45, 46, 49, 50, 57, 28, 59]. However, as it was discussed, first by
Pindzola et al. [31] and later eg. by Sato and Takeshi [108], a single
determinant wavefuction, whether constructed from Hartree-Fock or Kohn-Sham
spinorbitals, is inherently unable to describe the single ionization
processes, especially in closed-shell systems. This is due to the fact that if
two spinorbitals in the ground state share the same spatial component and
there is no spin-dependent perturbation in the time-dependent operator
representing the interaction between electrons and the external laser field
(as it is usually assumed), they will always evolve in the same manner in
order to maintain the closed-shell structure, making it impossible to
represent a state in which one electron becomes ionized and the other remains
in the ionic core. Since tunneling ionization is an intrinsic part of HHG and
may compete with above threshold and barrier suppression ionization under
certain laser conditions, this may give an incorrect physical picture of the
whole process. The way to overcome this issue is either to use a spin-
dependent external field that enforces different spatial movement of electrons
of different spin [109], or turn to multideterminant correlated methods like
time-dependent coupled cluster, time-dependent algebraic diagrammatic
construction (TD-ADC) [110, 111] and the above-mentioned TD-CISD.

In this work, we explore the applicability of the TD-CISD approach for the
calculation of the HHG spectra, aiming at answering the question whether the
explicit inclusion of the electronic correlation effects brings a substantial
improvement to the description of the harmonic generation process, that can
outweigh the cumbersome computational costs of this method. In order to do
this, we perform the real-time TD-CISD propagations of the helium atom
wavefunction, which serves us as a model multi-electron system. From the
propagations we compute the HHG spectra and compare them to analogous results
obtained using the TD-CIS method and the grid-based single-active-electron
calculations. The CI helium wavefunctions are expanded in single-electron
atom-centered Gaussian basis sets of different sizes. This also allows us to a
address another question of which way of enhancing the time-dependent
wavefunction expansion – by augmenting the configurational basis through the
addition of double excitations or by increasing the orbital basis by adding
more Gaussian functons – provides a better improvement to the description of
HHG. Choosing a two-electron atom has also an additional advantage, as for
such a system CISD is equivalent to full CI (no excitations higher than double
can be generated). This gives us the opportunity to compare results obtained
using a completely uncorrelated method with the ones from a method that
provides an exact correlation energy within a given basis set.

This paper is organised as follows. In section 2 we provide a brief
theoretical background of the applied methods and discuss the computational
details. Then in section 3 we present and analyze our results. Finally, in
section 4 we summarize our work and provide some outlook for future studies.

##  II Methods

###  II.1 The time-dependent configuration interaction formalism

We solve the time-dependent Schrödinger equation (TDSE) for the electronic
wavefunction of the helium atom subjected to intense laser field (the Hartree
atomic units are used throughout the paper unless stated otherwise)

$i\frac{\partial}{\partial
t}|\Psi(\mathbf{r},t)\rangle=(\hat{H}_{0}+\hat{V}(t))|\Psi(\mathbf{r},t)\rangle,$
(1)

where $\hat{H}_{0}$ is the time independent electronic Hamiltonian in the
Born-Oppenheimer approximation (assuming the position of the nucleus at the
center of the coordinate system),

$\hat{H}_{0}(\vec{r}_{1},\vec{r}_{2})=-\frac{\nabla_{1}^{2}}{2}-\frac{\nabla_{2}^{2}}{2}-\frac{2}{r_{1}}-\frac{2}{r_{2}}-\frac{1}{\lvert\vec{r}_{1}-\vec{r}_{2}\rvert},$
(2)

and $\hat{V}(t)$ is the time-dependent operator coupling the electrons to the
external field. Within the real-time TD-CI approach the time-dependent
wavefunction is written as a linear combination of the time-independent CI
states (appoximate eigenstates of $\hat{H}_{0}$)

$\ket{\Psi(t)}=\sum_{k}c_{k}(t)\ket{\Psi^{\textrm{CI}}_{k}}$ (3)

Each CI state can be expanded in the basis constisting of the ground state
(reference) Slater determinant $\Phi_{0}$ built from occupied spinorbitals
obtained through the solution of Hartree-Fock equations within a chosen basis
set, and the set of excited determinants,
$\\{\Phi_{\nu}^{\gamma},\Phi_{\nu\mu}^{\gamma\delta},...\\}$ that are created
from the reference determinant by replacing a certain number of occupied
spinorbitals $\\{\nu,\mu,...\\}$ with virtual spinorbitals
$\\{\gamma,\delta,...\\}$. The maximum number of replaced spinorbitals is one
for CIS states, two for CISD states, three for CISDT states etc. However, in
case of the real-time TD-CI calculations the wavefunction is bound to evolve
only on the manifold of states with spin equal to the ground state spin, so it
is more convenient to express the CI states in the basis of configuration
state functions (CSFs) – linear combinations of Slater determinants, which are
eigenfunctions of the square of the total spin operator. This allows for a
significant reduction of the necessary CI space. For a closed-shell system,
such as the helium atom, besides the HF reference determinant, we have one
type of CIS singlet excited CSFs (the roman letters denote the spatial
orbitals, and the bars indicate whether we replace an $\alpha$ or a $\beta$
spinorbital associated with a given orbital) [112, 64],

${}^{\mathrm{1}}\Phi_{i}^{a}=\frac{1}{\sqrt{2}}(\Psi_{i}^{a}+\Psi_{\bar{i}}^{\bar{a}}),$
(4)

and five types of CISD singlet excited CSFs,

${}^{\mathrm{1}}\Phi_{ii}^{aa}=\Phi_{i\bar{i}}^{a\bar{a}},$
${}^{\mathrm{1}}\Phi_{ii}^{ab}=\frac{1}{\sqrt{2}}(\Phi_{i\bar{i}}^{a\bar{b}}+\Phi_{\bar{i}i}^{\bar{a}b}),$
${}^{\mathrm{1}}\Phi_{ij}^{aa}=\frac{1}{\sqrt{2}}(\Phi_{i\bar{j}}^{a\bar{a}}+\Phi_{\bar{i}j}^{\bar{a}a}),$
(5)
${}^{\mathrm{A}}\Phi_{ij}^{ab}=\frac{1}{\sqrt{12}}(2\Phi_{ij}^{ab}+2\Phi_{\bar{i}\bar{j}}^{\bar{a}\bar{b}}+\Phi_{i\bar{j}}^{a\bar{b}}+\Phi_{\bar{i}j}^{\bar{a}b}-\Phi_{i\bar{j}}^{b\bar{a}}-\Phi_{\bar{i}j}^{\bar{b}a}),$
${}^{\mathrm{B}}\Phi_{ij}^{ab}=\frac{1}{2}(\Phi_{i\bar{j}}^{a\bar{b}}+\Phi_{\bar{i}j}^{\bar{a}b}+\Phi_{i\bar{j}}^{b\bar{a}}+\Phi_{\bar{i}j}^{\bar{b}a}),$

where $i>j$, $a>b$, so the total expressions for CIS and CISD states read

$\displaystyle\Psi^{\textrm{CIS}}_{k}=$ $\displaystyle
C_{0,k}\Phi_{0}+\sum_{i,a}C^{a}_{i,k}{{}^{\mathrm{1}}}\Phi_{i}^{a},$ (6)
$\displaystyle\Psi^{\textrm{CISD}}_{k}=$ $\displaystyle
C_{0,k}\Phi_{0}+\sum_{i,a}(C^{a}_{i,k}{{}^{\mathrm{1}}}\Phi_{i}^{a}+C^{aa}_{ii,k}{{}^{\mathrm{1}}}\Phi_{ii}^{aa})$
(7)
$\displaystyle+\sum_{i,a,b}C^{ab}_{ii,k}{{}^{\mathrm{1}}}\Phi_{ii}^{ab}+\sum_{i,j,a}C^{aa}_{ij,k}{{}^{\mathrm{1}}}\Phi_{ij}^{aa}$
$\displaystyle+\sum_{i,j,a,b}({{}^{\mathrm{A}}}C^{ab}_{ij,k}{{}^{\mathrm{A}}}\Phi_{ii}^{ab}+{{}^{\mathrm{B}}}C^{ab}_{ij,k}{{}^{\mathrm{B}}}\Phi_{ii}^{ab}).$

These are general expressions valid for an arbitrary closed-shell system,
however is is worth noting that in case of the helium atom the CSF types
$\Phi_{ij}^{aa}$, ${}^{A}\Phi_{ij}^{ab}$ and ${}^{B}\Phi_{ij}^{ab}$ are absent
from the second expansion, since there is only one occupied spatial orbital in
the HF ground state. The coefficients $C_{k}$ and the eigenenergies of
corresponding CI states $E_{k}^{\textrm{CI}}$ are determined by diagonalizing
the matrix of the $H_{0}$ operator in the basis of all CSFs available at the
given level of theory,

$\mathbf{H_{0}^{\textrm{CI}}}\mathbf{C}_{k}=E_{k}^{\textrm{CI}}\mathbf{C}_{k}$
(8)

The expressions for the matrix elements of $\mathbf{H_{0}^{\textrm{CIS}}}$ and
$\mathbf{H_{0}^{\textrm{CISD}}}$ are provided in the supplementary material.

###  II.2 Real-time propagations

In our calculations the laser pulse is represented by a linearly polarized
oscillating electric field with a sine-squared envelope of shape

$E(t)=\begin{cases}E_{0}\sin(\omega_{0}t)\sin^{2}(\omega_{0}t/2n_{c})&\text{if}\;0\leq
t\leq 2\pi n_{c}/\omega_{0},\\\ 0&\text{otherwise,}\end{cases}$ (9)

with the carrier frequency $\omega_{0}$ = 1.55 eV ($\lambda_{0}$ = 800 nm),
the number of optical cycles $n_{c}$ = 20, and the total pulse duration $2\pi
n_{c}/\omega_{0}\approx$ 2206 a.u. ($\approx$ 53.4 fs). The maximum field
amplitude $E_{0}$ is related to the laser intensity $I_{0}$ via
$I_{0}=\varepsilon_{0}cE_{0}^{2}/2$. We examine three laser intensities:
$2\times 10^{14}$ W/cm2, $3\times 10^{14}$ W/cm2 and $5\times 10^{14}$ W/cm2.
Their choice was based mainly on the value of the Keldysh parameter
$\gamma=\sqrt{I_{p}/2U_{p}}$ [113], where $I_{p}$ is the first ionization
energy of helium and $U_{p}=E_{0}^{2}/4\omega_{0}^{2}$ is the ponderomotive
energy of a free electron in the electromagnetic field. For the investigated
laser intensities the Keldysh parameter equals 1.01, 0.83 and 0.64,
respectively, indicating the tunneling regime of ionization, which is
desirable for an effective harmonic generation process.

The external electric field is treated in the dipole approximation and the
field coupling operator $V(t)$ is expressed in the length gauge (assuming that
the field is polarized along the $z$-axis),

$\hat{V}(t)=-\hat{z}E(t).$ (10)

The wavefunction is propagated using the middle-step Crank-Nicolson algorithm,

$\displaystyle\left(\mathbf{I}+\frac{i\Delta
t}{2}\mathbf{H^{\textrm{CI}}}(t+\Delta t/2)\right)\mathbf{c}(t+\Delta t)=$
$\displaystyle=\left(\mathbf{I}-\frac{i\Delta
t}{2}\mathbf{H^{\textrm{CI}}}(t+\Delta t/2)\right)\mathbf{c}(t),$ (11)

where $\mathbf{H^{\textrm{CI}}}(t)$ is the time-dependent Hamiltonian matrix
(and CI refers to either CIS or CISD), $\mathbf{I}$ is the identity matrix and
$\mathbf{c}(t)$ is the vector of the time-dependent coefficients $c_{k}(t)$.
Since the time-dependent wavefunction is expanded in the basis of CI
eigenstates, the time-dependent Hamiltonian matrix simplifies to
$\mathbf{H^{\textrm{CI}}}(t)$ = $\mathbf{E^{\textrm{CI}}}+\mathbf{V}(t)$,
where $\mathbf{E^{\textrm{CI}}}$ is the diagonal matrix of CI eigenenergies.
We start the propagation from the ground state of helium (calculated at either
CIS or CISD level of theory, depending on the propagation type) and propagate
the system in time for twice the duration of the laser pulse, with timestep
$\Delta t$ = 0.01 a.u.

###  II.3 Gaussian basis sets

Difficulties in the proper representation of Rydberg and continuum electronic
states are the major drawback of modeling the laser-driven electron dynamics
using the basis set approach. These states are usually highly diffuse, so
approximating them with a set of nuclei-centered functions may be troublesome.
Some recent works, however, have shown [114, 115, 116, 117] that a good semi-
quantitative description of HHG can be achieved when using a special type of
Gaussian basis sets consisting of two types of functions. The first type is
some standard basis set routinely used in quantum chemical calculations of
ground-state properties, such as one of the Dunning basis sets [118]. It
comprises functions that can well reproduce the electronic ground state and
some of the lowest-lying excited states, but fails to depict the motion of
electrons far from the nuclei. Therefore it must be supplemented with the
second type of functions designed to describe the highly excited and continuum
states. Several types of Gaussian functions have been proposed for this second
type, including functions developed by Kaufmann et al. [119, 115, 116, 117]
and even-tempered Gaussians [79, 83].

In our recent paper [120] we introduced a novel class of Gaussian functions we
dubbed ARO functions, that proved able to approximate a wide spectrum of
excited states necessary for the description of HHG in both one- and two-
electron systems. Therefore, in the present calculations we use Dunning basis
sets mixed with ARO functions. In order to select an optimal basis set for the
time-dependent calculations we have performed a series of test propagations of
the helium atom using different augmented correlation-consistent basis sets of
general type x-aug-cc-pVYZ, with x=$\emptyset$,d,3,4 and Y=D,T,Q,5,
supplemented with shells of ARO functions with angular momenta $l$ from 0
($s$-type functions) up to $l_{\textrm{max}}$ = 2, 3 or 4 ($d$-,$f$\- and
$g$-type functions, respectively). The ARO functions were generated according
to the procedure described in [120], by fitting Gaussians to a set of Slater-
type orbitals with exponent $\zeta$ = 1.7 (the effective nuclear charge of
helium) and principal quantum numbers $n$ from 2 to 90. The number of ARO
fuctions added per angular momentum varied from 4 to 10. The test calculations
were performed at the TD-CIS level of theory, for all examined laser
conditions. From the results of the propagations the HHG spectra were computed
using the methodology described below and assessed by comparing their key
features, including the distinguishability of harmonic peaks, the cutoff
position and presence of numerical artifacts. The properties of the basis sets
themselves, such as lack of linear dependencies between the functions, were
also taken into account. Overall, the most reliable results were obtained by
using basis sets based on the d-aug-cc-pVQZ Dunning basis set, out of which
two were selected for the proper TD-CI calculations. The first one includes 8
ARO functions per each angular momentum from 0 to 4 and is dubbed d-aug-cc-
pVQZ+ARO8g. It provided the most optimal spectra at the TD-CIS level, however
its size – 262 functions in total – makes it too large to use in the TD-CISD
calculations (the dimension of the CIS matrix expressed in singlet CSFs is
$ov+1$, where $o$ and $v$ are the numbers of occupied and virtual orbitals,
respectively, while the dimension of the CISD matrix is
$\frac{1}{2}(o^{2}v^{2}+3ov+2)$). This is why we also use its smaller variant
containing 6 ARO functions per each angular momentum from 0 to 3 (158
functions in total, named d-aug-cc-pVQZ+ARO6f), in which we perform both the
TD-CIS and TD-CISD calculations.

We are aware of the fact that ideally the basis set should be optimized (in
terms of number and range of the ARO functions) separately for each laser
intensity, since fields of different strength lead to different population of
electronic states. However, since this paper focuses more on exploring new
methods for simulating HHG than the construction of basis sets, we decided to
use the same basis sets for all laser conditions for the sake of simplicity.

The one- and two-electron integrals necessary for the calculations are
generated using the Dalton2020.0 software [121]. All other calculations,
including solving the Hartree-Fock equations, construction and diagonalization
of the CI matrices and real-time propagations are performed using a home-made
program.

###  II.4 Treatment of ionization within the TD-CI approach

The incompleteness of the used basis sets, both in terms of the radial
functions and angular momenta, causes the wavefunction to undergo unphysical
reflections when it reaches the area where it cannot be properly described
with Gaussian functions. A similar problem is encountered in the grid-based
calculations, where the wavefunction is being reflected after reaching the
grid boundary. Several methods have been developed to compensate for such an
unwanted behavior, including complex absorbing potentials [122, 123], exterior
complex scaling [124] and mask functions [125]. All of them rely on
eliminating these parts of the wavefuction that cannot be reproduced by the
propagation method in use, effectively simulating the ionization process. In
our calculations we implement the so-called heuristic lifetime model developed
by Klinkusch et al. [67] specifically for the TD-CI propagation scheme. The
model relies on the assumption that the CI states above the ionization
threshold have finite lifetimes and replaces their energies by complex
counterparts,

$E_{k}^{\textrm{CI}}\rightarrow
E_{k}^{\textrm{CI}}-\frac{i}{2}\Gamma_{k}\quad\text{for}\;E_{k}^{\textrm{CI}}\geq
E_{0}^{\textrm{CI}}+I_{p},$ (12)

where $\Gamma_{k}$ is the ionization rate (inverse lifetime) of the $k$-th CI
state. The values of $\Gamma_{k}$ are calculated in the one-electron picture.
Assuming that an electron on orbital $a$ with positive energy $\epsilon_{a}$
has the kinetic energy $\epsilon_{a}=\frac{1}{2}v^{2}$, one can introduce an
empirical parameter $d_{a}$ defining the distance the electron can travel
before being ionized, and derive the ionization rate $\gamma_{a}$ of this
orbital as its inverse lifetime $\tau_{a}$,

$\gamma_{a}=1/\tau_{a}=\theta(\epsilon_{a})\sqrt{2\epsilon_{a}}/d_{a},$ (13)

where $\theta(x)$ is the Heaviside step function. Therefore the expressions
for the ionization rates of CIS and CISD states read

$\displaystyle\Gamma_{k}^{\textrm{CIS}}=$
$\displaystyle\sum_{i,a}\absolutevalue{C^{a}_{i,k}}^{2}\gamma_{a},$ (14)
$\displaystyle\Gamma_{k}^{\textrm{CISD}}=$
$\displaystyle\sum_{i,a}(\absolutevalue{C^{a}_{i,k}}^{2}\gamma_{a}+\absolutevalue{C^{aa}_{ii,k}}^{2}2\gamma_{a})$
$\displaystyle+\sum_{i,a,b}\absolutevalue{C^{ab}_{ii,k}}^{2}(\gamma_{a}+\gamma_{b})+\sum_{i,j,a}\absolutevalue{C^{aa}_{ij,k}}^{2}2\gamma_{a}$
$\displaystyle+\sum_{i,j,a,b}(\absolutevalue{{{}^{\mathrm{A}}}C^{ab}_{ij,k}}^{2}(\gamma_{a}+\gamma_{b})+\absolutevalue{{{}^{\mathrm{B}}}C^{ab}_{ij,k}}^{2}(\gamma_{a}+\gamma_{b})).$

Klinkusch et al. originally considered only one value of $d$ for all orbitals
with positive $\epsilon_{a}$ [67]. However, Coccia et al. [115] recently
reported that, due to a relatively small number of continuum electronic states
obtained with Gaussian basis sets and an uneven spacing between their energy
levels, such a choice leads to numerous artifacts in the HHG spectra, like an
incorrect description of the harmonic cutoff. Thus, in our version of the
model we use two values of $d$. For orbitals with energy smaller than the
harmonic cutoff energy predicted by the three-step model,
$E_{\textrm{cut}}=I_{p}+3.17U_{p}$, we use $d_{1}$ equal to the semiclassical
electron quiver amplitude in an electromagnetic field, $E_{0}/\omega_{0}^{2}$,
while for orbitals with energy exceeding $E_{\textrm{cut}}$ we use $d_{2}$ =
0.1. The value of $I_{p}$ is obtained in the Hartree-Fock approximation from
the Koopmans’ theorem.

It should be noted that our correction to the finite lifetime model is
somewhat similar to the correction proposed by Coccia et al. [115, 84], who
assigned different values of $d$ to whole CI states based on their total
energy, rather than to single orbitals. The two approaches are essentially the
same for a single-electron system, and give similar ionization rates of CIS
states, but the ionization rates of CISD states provided by them may vary
significantly. For instance we can consider a state dominated by
configurations with double excitations to orbitals with energies just above
zero. Since states with a large contribution of doubly excited configurations
are usually characterized by high $E_{k}^{\textrm{CISD}}$, it may occur that
the total energy of this state exceeds $E_{\textrm{cut}}$ and this state is
classified by the model of Coccia et al. as having an unphysically short
lifetime, even though the kinetic energies of both excited electrons are
relatively low. Our version of the model is more consistent with the single-
electron picture adopted by Klinkusch et al., since it acknowledges that it’s
the kinetic energy of single electrons what limits the harmonic generation and
not the total energy of $N$-electron wavefunctions.

###  II.5 Grid propagations

Aside from the basis set propagations, we also solve the time-dependent
Schrödinger equation for the helium atom on a spatial grid, at the same laser
conditions. Since the direct propagation of the six-dimensional wavefunction
is currently out of range, we perform the calculations within the single-
active-electron (SAE) approximation. In terms of the depiction of the
electronic wavefunction this is the crudest method of all applied in this
work, as not only it completely neglects all correlation effect, but it also
explicitly considers the motion of only one electron, while the other is
represented together with the nucleus in the form of a spherically symmetric
effective potential. On the other hand, the SAE approximation is known to well
reproduce the key features of the HHG spectra [126, 127]. What is even more
important, grid projection of the wavefunction provides a very close
approximation to the complete orbital basis, which is why the SAE approach is
often used as a benchmark for basis set calculations [45, 57, 59].

We perform the grid-based propagations using a modified version of QPROP TDSE
solver [128, 129]. The SAE wavefunction is expanded into partial waves with
angular momenta up to $l$=50. The angular dependence of each partial wave is
treated analytically using spherical harmonics, while its radial component is
discretized on a one-dimensional grid with spacing $\Delta r$=0.1 a.u. The
time-dependent Hamiltonian reads

$\hat{H}(\vec{r},t)=-\frac{\nabla^{2}}{2}+V_{\textrm{SAE}}(r)+V_{\textrm{CAP}}-iA(t)\frac{\partial}{\partial
z},$ (16)

where the last term is the electron-field interaction operator expressed in
velocity gauge through the vector potential A(t) connected to the time-
dependent electric field via $E(t)=-\partial_{t}A(t)$. $V_{\textrm{SAE}}$ is
the SAE potential,

$V_{\textrm{SAE}}(r)=-\frac{1+e^{-2.135r}}{r},$ (17)

that is known to approximately reproduce the first few ground and excited
state energies of the helium atom. $V_{\textrm{CAP}}$ is the complex absorbing
potential (CAP) that simulates the ionization process in a similar manner to
the finite lifetime model used in TD-CI calculations. We use CAP in the form
developed by Manolopoulos et al. [130], with absorption parameters
$\delta=0.2$ and $k_{\mathrm{min}}=0.2$. For the consistency with TD-CI
propagations the CAP starting position is also placed at the distance from the
nucleus equal to the electron quiver amplitude in the laser field.

###  II.6 Calculation of the HHG spectra

After performing real-time propagations we calculate the HHG spectra in the
dipole form and in the velocity form, using the Fourier transform,

$I_{\mathrm{HHG}}(\omega)=\absolutevalue{\frac{1}{t_{f}-t_{i}}\int_{t_{i}}^{t_{f}}G(t)e^{i\omega
t}dt}^{2},$ (18)

where $t_{i}$ and $t_{f}$ are initial and final propagation times,
respectively, and $G(t)$ is either the time-resolved dipole moment calculated
in the TD-CI picture as

$\expectationvalue{D(t)}=\sum_{k,l}[c_{l}(t)]^{\dagger}c_{k}(t)\langle\Psi_{l}|\hat{z}|\Psi_{k}\rangle,$
(19)

or the dipole velocity, computed by the virtue of the Ehrenfest theorem,

$\expectationvalue{\dot{D}(t)}=-i\sum_{k,l}[c_{l}(t)]^{\dagger}c_{k}(t)\langle\Psi_{l}|\hat{\partial}_{z}|\Psi_{k}\rangle.$
(20)

In case of grid propagations, QPROP does only provide the time-resolved dipole
moment and not the time-resolved dipole velocity. Therefore we compute the
velocity form of the spectra using the approximate relation between the dipole
form $I_{\mathrm{HHG,z}}(\omega)$ and the velocity form
$I_{\mathrm{HHG,v_{z}}}(\omega)$:

$I_{\mathrm{HHG,v_{z}}}(\omega)\approx\omega^{2}I_{\mathrm{HHG,z}}(\omega).$
(21)

##  III Results

Let us start the analysis of the obtained results with some general
consideration regarding how the inclusion of double excitations in the CI
picture affects the obtained time-independent Hamiltonian eigenspectrum, as it
will help us better understand the features of the HHG spectra. We can
distinguish three effects stemming from adding doubly excited configurations
to the CI wavefunction. The first one is the increase in the total number of
states, which is rather self-explanatory given the larger size of the CISD
Hamiltonian matrix. As mentioned in the preceding section, CISD states
dominated by doubly excited configurations usually have high energies, so one
can expect a denser distribution and wider range of states mainly above the
ionization threshold. The second effect is the lowering of energies of the CI
states that are already approximated by the CIS level of theory (including the
ground state) by the values of correlation energy. The third and final effect
is the ability of CISD to reproduce states of higher total angular momentum
than CIS within the same basis set. This is due to the total angular momentum
of a many-electron wavefunction $L$ being a sum of angular momenta of single
electrons. In case of the helium atom, CIS incorporates only the
configurations in which one electron is excited to an orbital with any $l$ up
to $l_{\textrm{max}}$, while the other remains on the ground state orbital
with $l$=0, thus the maximum total angular momentum of the two-electron
wavefunction is equal to $l_{\textrm{max}}$. In CISD on the other hand both
electrons can be potentially excited to orbitals with $l$=$l_{\textrm{max}}$,
so the maximum total angular momentum equals $2l_{\textrm{max}}$. Therefore on
the CISD/d-aug-cc-pVQZ+ARO6f level of theory we can obtain states with higher
total angular momentum than on the CIS/d-aug-cc-pVQZ+ARO8g level of theory,
despite using basis set with less orbital angular momenta. These three effects
obviously cannot be strictly separated from each other, but their mutual
interplay governs the description of the laser-driven electron dynamics.

Figure 1: Distributions of energies and ionization probabilities of the CI
states obtained at the CIS/d-aug-cc-pVQZ+ARO6f, CIS/d-aug-cc-pVQZ+ARO8g and
CISD/d-aug-cc-pVQZ+ARO6f level of theory. Inset on the bottom plot shows the
distribution for the first 160 CISD states, for an easier comparison with two
upper plots

The energies and ionization rates of CI states computed at the CIS/d-aug-cc-
pVQZ+ARO6f, CIS/d-aug-cc-pVQZ+ARO8g and CISD/d-aug-cc-pVQZ+ARO6f levels of
theory (hereafter CIS/6f, CIS/8g and CISD/6f for short, respectively) are
plotted on Fig. 1. As predicted, both CIS/6f and CISD/6f calculations provide
a similar number of about 60 bound states, while the number of high-energy
continuum states in the former method is smaller by two orders of magnitude
than in the latter. Contrariwise, the addition of more ARO functions to the
basis set results in much higher density of states close to the ionization
threshold. The ionization rates of CIS states calculated in both basis sets
rise consecutively with the state number. There is a sharp increase in
$\gamma$ about 105th and 190th state in CIS/6f and CIS/8g eigenspectrum,
respectively, indicating that the states beyond these points contain
excitations to orbitals with $\epsilon>E_{\textrm{cut}}$. The same dependency
for the CISD states is far more oscillatory, which illustrates the effect
discussed in the previous section: states dominated by configurations in which
one electron is excited to a high-energy continuum orbital and the other is
excited to a bound orbital or not excited at all (e.g. states with high total
ionization rates), and states dominated by configurations with double
excitations to low-energy continuum orbitals (e.g. states with low total
ionization rates) may have similar total energies.

The HHG spectra obtained from the real-time propagations are presented on Fig.
2. We start the comparison with some remarks on the grid-based results. The
SAE approximation is shown to well reproduce the overall shape of the spectra
in both forms and at all laser conditions. The intensities of the consecutive
peaks in the harmonic plateaux decrease slowly and almost monotonically with
the harmonic number. Each plateau ends with a sharp cutoff, after which the
HHG intensity abruptly drops by a several orders of magnitude. The cutoff
position is in a good agreement with the value predicted by the three-step
model at every examined laser intensity. On the other hand, the harmonic peaks
are less distinct and more contaminated by numerical noises than the ones on
the spectra obtained from TD-CI propagations. The intensity of these noises
rises with the harmonic number, so that at the spectrum corresponding to
intensity $5\times 10^{14}$ W/cm2 the peaks before the cutoff cannot be
distinguished, suggesting a rather poor representation of highly excited and
continuum states in the SAE picture. It should be noted that these artifacts
could not be eliminated by adding more partial waves to the wavefunction
expansion or by decreasing the grid spacing, implying the limitations of the
SAE model itself.

At the laser intensity $2\times 10^{14}$ W/cm2 all three TD-CI methods perform
fairly good at reproducing the harmonic peaks in the plateau, as well as at
predicting the cutoff positon. However, on deeper analysis, TD-CISD/6f
produces the spectrum of arguably the best quality. It provides the smallest
amount of excess harmonic peaks behind the cutoff and the least numerical
noises in the harmonic background. Moreover, the plateaux on both TD-CIS-based
spectra are characterized by a triangular envelope, with the intensities of
the peaks increasing up to the point corresponding to the ionization
threshold, and then decreasing until reaching the cutoff. The TD-CISD-based
spectrum provides a more constant intensity of the plateau, resembling that in
the SAE-based spectrum. This effect is more noticeable in the velocity form.

Figure 2: HHG spectra of the helium atom at three examined laser intensities,
in the dipole form (top row) and in the velocity form (bottom row). The TD-
CIS/6f, TD-CIS/8g and TD-CISD/6f spectra are upshifted along the $y$-axis by
the values of 2, 4 and 6, respectively, for sake of clarity. The theoretical
cutoff position predicted by the three-step model (solid line) and the
harmonic corresponding to the ionization energy (dashed line) are denoted on
each plot

At the laser intensity $3\times 10^{14}$ W/cm2 TD-CIS/6f is still able to
reproduce most of the harmonic plateau, although its performance starts to
decline, with the cutoff harmonic peak being just distinguishable in the
dipole form of the TD-CIS/6f spectrum, and the cutoff position being placed
beyond the value predicted by the three-step model in the velocity form. A
much better description of the cutoff region is achieved with TD-CIS/8g,
although at the cost of some numerical artifacts in the background, visible
particularly in the velocity form of the spectrum.

Surprisingly, TD-CISD/6f provides a poorer description of the HHG plateau at
this intensity than both TD-CIS/6f and TD-CIS/8g, with the peaks being less
distinct especially in the region between about the 40th harmonic and the
cutoff. A potential explanation for this observation is following. On the
basis of the differences between the TD-CIS/6f and TD-CIS/8g spectrum we can
infer that the states with $L>3$ are necessary to better reproduce the highest
order peaks in the plateau. In CISD/6f such states can be constructed, but
only from configurations with double excitations to orbitals with $l\leq 3$.
However, the ARO functions are optimized in the one-electron picture, so it is
natural that an ARO shell with angular momentum $l$ is best suited for
describing states with only one electron excited to orbital with the same
angular momentum. Therefore, it is inevitable that the properties of states
with $L>3$, composed solely of doubly excited configurations, are approximated
less accurately than the properties of states with $L\leq 3$, comprised of
both singly and doubly excited configurations. This applies in particular to
the dipole moment expectation value, from which the HHG spectrum is derived.
This imbalanced description of different angular momenta within the time-
dependent wavefunction is a source of additional artifacts, which do not occur
if the states with $L>3$ are not included at all. On the other hand, it is
worth noticing that of all TD-CI-based spectra in the dipole form a decrease
of background intensity beyond the cutoff is present only in the TD-CISD/6f
one. Given that such a decrease is observed in both forms of all grid-based
spectra, this suggests a possible key role of states with high angular momenta
in the depiction of this feature.

At the laser intensity $5\times 10^{14}$ W/cm2 the performance of TD-CIS/6f
breaks down, as the HHG intensity abruptly falls off after the 63th harmonic
and no higher order peaks are present in the spectrum. This indicates the lack
of CIS states with energy or angular momentum high enough to describe the
laser-driven electron dynamics in such an intense field. TD-CIS/8g on the
other hand is still able to reproduce all harmonic peaks in the plateau at
this intensity, with the performance comparable to that of TD-CIS/6f at
intensity $3\times 10^{14}$ W/cm2. TD-CISD/6f provides a rather similar
picture to TD-CIS/6f, with only minor differences. In the first half of the
plateau, encompassing harmonics from first to 40th, we observe some
improvement in terms of more constant intensity of consecutive peaks than in
both TD-CIS-based spectra. This effect is similar to the one observed at
intensity $2\times 10^{14}$ W/cm2, and is a strong hint for the role of
electronic correlation in HHG. This is because these lower order peaks are
generated mostly by transitions to states with relatively low excitation
energies, which are approximately described by all three levels of theory, but
in TD-CISD/6f this description is corrected by including doubly excited
configurations. In the second half of the plateau (region between the 40th
harmonic and the cutoff) the peaks that are visible clearly in the TD-CIS/6f
spectrum are less distinct here, due to the same reason as in case of the
intensity $3\times 10^{14}$ W/cm2. However, it can be noticed that the region
between the 63th harmonic and the cutoff the HHG intensity is slightly
uplifted compared to the TD-CIS/6f spectrum, indicating transitions to states
with $L>3$, that are impossible to realize on the TD-CIS/6f level of theory.
Nevertheless, the poor description of these states using only doubly excited
configurations makes it unfeasible to distinguish individual harmonic peaks.

##  IV Conclusion

In this work we have compared the performance of three real-time time-
dependent wavefunction propagation methods: TD-CI with single excitations, TD-
CI with single and double excitations, and the single-active-electron
approximation, in the modeling of high harmonic generation in the helium atom
exposed to laser fields of extreme intensity. The first two approached rely on
expanding the time-dependent wavefunction in the predefined basis set
comprised of Gaussian functions, while the third is grid-based. The full HHG
spectra obtained from TD-CISD calculations are to our knowledge the first to
be reported in the literature.

First of all, the results presented in this paper may serve as an additional
proof of applicability of the ARO Gaussian functions in the modeling of HHG,
extending our previous study [120]. We have shown on the example of the helium
atom that Dunning basis sets supplemented with ARO functions allow for
studying the laser-driven electron dynamics in many-electron systems in the CI
framework. Also, it has been proven that subsequent addition of ARO shells
with higher angular momenta, as well as augmenting the number of functions per
angular momentum makes it possible to simulate the optical response in laser
fields of increasing intensity.

This work was mainly motivated by the question of whether the explicit
inclusion of electron correlation via the incorporation of doubly excited
configurations in the time-dependent CI wavefunction has a significant
contribution to the description of HHG. After a careful analysis of the
obtained HHG spectra we are able to reach the conclusion that the addition of
double excitations can indeed noticeably improve the TD-CI picture of HHG, but
only when the basis set in which the calculations are performed already
provides qualitatively acceptable results at the TD-CIS level of theory. This
is best seen at the intensity $2\times 10^{14}$ W/cm2. The eigenspectrum of
CIS/6f already contains all the singly excited states necessary for an
approximate description of the laser-driven wavefunction, and the role of
double excitations in CISD/6f is merely to complement this description by
including the correlation between the electron that becomes ionized and the
one that remains in the residual ion. The effects of this complementation are
perceptible in both an improved overall shape of the HHG plateau and a better
description of the cutoff region, and are an indisputable evidence for the
contribution of electronic correlation in the HHG process. On the other hand,
when the basis set lacks the functions of angular momenta high enough to
describe the electron dynamics, augmenting the configurational basis by
including double excitations cannot compensate this shortcoming and the
correct picture can only be regained by expanding the orbital basis, as seen
at higher laser intensities. This is because the states exclusive to the CISD
eigenspectrum are better suited to describe double excitations and double
ionization processes, while in HHG the contribution of single ionization
obviously prevails over the contribution of double ionization. Thus,
performing TD-CISD calculations in an inadequate basis set brings little to no
improvement over the TD-CIS picture (as in the case of intensity $5\times
10^{14}$ W/cm2) or can even worsen it (as in the case of intensity $3\times
10^{14}$ W/cm2). Moreover, although CISD provides a far higher overall number
of CI states, the majority of them lay far above the ionization threshold, so
their presence is not expected to contribute much to the HHG process. On the
contrary, expanding the number of ARO functions in the basis set leads to
significant increase in the number of states just near the ionization
threshold, which are believed to play a major role in HHG.

Unfortunately, such a conclusion limits the applicability of the TD-CISD
approach to systems for which non-correlated calculations in a comparatively
small basis set already provide a qualitatively correct results, i.e. atoms or
molecules with few electrons, subjected to fields of relatively low intensity.
Otherwise, incrementing the number of electrons or the laser intensity means
that larger basis sets have to be used in order to capture the electronic
correlation effects, which in turn makes the TD-CISD calculations impractical
due to their high computational complexity. There are however some potential
ways to bypass this restriction. For example, one can truncate the CISD
excitation space by removing the highest lying eigenstates, which have
negligible contribution to the electron dynamics. Another solution may be to
perform calculations for larger atoms or molecules using the effective core
potential method, which allows for limiting the number of explicitly treated
electrons to valence shell only. Both approaches significantly reduce the
order of time propagation matrix equations and will be explored in future
works.

This paper also introduces and test a new correction to the finite lifetime
ionizaton model of Klinkusch et al. [67]. As shown by our results, it ensures
a fairly good agreement of the cutoff position with the value predicted by the
three-step model even at extremely high laser intensities.

## Acknowledgments

We acknowledge support from the National Science Centre, Poland (Symfonia
Grant No. 2016/20/W/ST4/00314). M. L. also acknowledges support from ERC AdG
NOQIA; Agencia Estatal de Investigación (R&D project CEX2019-000910-S, funded
by MCIN/AEI/10.13039/501100011033, Plan National FIDEUA PID2019-106901GB-I00,
FPI, QUANTERA MAQS PCI2019-111828-2, Proyectos de I+D+I “Retos Colaboración”
RTC2019-007196-7); Fundació Cellex; Fundació Mir-Puig; Generalitat de
Catalunya through the CERCA program, AGAUR Grant No. 2017 SGR 134,
QuantumCAT\U16-011424, co-funded by ERDF Operational Program of Catalonia
2014-2020; EU Horizon 2020 FET-OPEN OPTOLogic (Grant No 899794); Marie
Skłodowska-Curie grant STREDCH No 101029393; “La Caixa” Junior Leaders
fellowships (ID100010434) and EU Horizon 2020 under Marie Skłodowska-Curie
grant agreement No 847648 (LCF/BQ/PI19/11690013, LCF/BQ/PI20/11760031,
LCF/BQ/PR20/11770012, LCF/BQ/PR21/11840013).

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