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ezDyson user’s manualA C++ program that calculates polyatomic photodetachment and

photoionization cross sections using Dyson orbitals

by

Liang Tao, Melania Oana and Anna Krylov

Department of Chemistry

University of Southern California

Los Angeles, CA 90089-0482

Telephone: (213) 740-4929

Email: [email protected] (Liang Tao)

[email protected] (Melania Oana)

[email protected] (Anna Krylov)

Website: http://iopenshell.usc.edu/downloads/ezdyson/

Work conducted in the framework of the iOpenShell Center

Contents

I. About ezDyson 2

II. Theoretical background 3

III. Input file 5

A. Generating input files 5

B. Input file structure 5

IV. Running ezDyson 15

A. Installation 15

B. Running the program 16

V. Output 16

A. Main output 16

B. Additional output files 18

VI. Examples 18

A. Photoionization 18

1. H2O 18

2. CO 21

3. CH4 23

4. CH3OH 25

5. CH2O 26

6. C2H4 photoionization 28

B. Photodetachment 29

1. H! 30

2. Li! 30

3. C! 31

A. Electric dipole interaction operator in laboratory frame 31

B. Total cross sections for isotropic averaging molecular orientations 34

References 35

2

I. ABOUT ezDyson

The program is available for download at iopenshell.usc.edu/downloads/

Please contact Prof. Anna I. Krylov ([email protected]), C. Melania Oana ([email protected])

or Liang Tao ([email protected]) if you have questions, problems, or bugs to report.

ezDyson is a C++ code that calculates absolute photodetachment/photoionization cross

sections, photoelectron angular distributions and beta anisotropy parameters using Dyson

orbitals computed by an ab initio program. The model used in the calculations is based on

the following approximations: (1) weak field limit, i.e., the photodetachment or photoioniza-

tion is teated in a perturbative regime; (2) dipole approximation i.e., the wavelength of the

radiation field is longer than size of the molecule, and electric dipole interaction is su!cient

to describe this one photon process; (3) sudden approximation, i.e., we ignore the interac-

tions between the photoelectron and the remaining core electrons; (4) strong orthogonality

condition, i.e., the continuum orbital is orthogonal to all the molecular orbitals contributing

to the initial state; (5) central potential, i.e., we use either the plane wave or Coulomb wave

to describe the continuum state of the photoelectron. The non-spherical molecular Coulomb

potential is approximated by the single-center Coulomb interaction (e-H+).

The code requires a standard input from quantum chemical calculations (geometries,

atomic basis set), details of the experiment (laser polarization, ionization energy, molecular

averaging, electron kinetic energies), and the coe!cients of the Dyson orbitals in the AO

basis and their norms. The ezDyson input is in xml format, and it can be prepared manually.

Alternatively, a Python interface can be used to extract data from an electronic structure

program. At this point, a Python interface is provided only for the Q-Chem outputs. Dyson

orbitals calculations are currently implemented within the EOM-CC suite of methods in Q-

Chem[3].

This manual provides a detailed description on the input structure and gives a number of

examples. It also presents a derivation of the working expressions using a time-independent

treatment of the photodetachment/photoionizaion based on Dyson orbital and angular mo-

mentum theory. The structure of the text is as follows: Section 1: brief explanations of the

model and working expressions; Section 2: description of the input and how to use it; Section

3: working examples; Appendix: detailed derivation using angular momentum theory. In

this new version 3.0, we added the analytical averaging method to calculate isotropic cross

3

sections and fixed a few minor bugs.

II. THEORETICAL BACKGROUND

Dyson orbitals are defined as the overlap between an N electron molecular wavefunction

and the N-1 electron wavefunction of the ionized system.

!d(1) =!

N

!

"NI (1, . . . , n)"N!1

F (2, . . . , n)d2 . . . dn (1)

They contain all the information about the ionized system for the cross-section calculation:

d"

d#k=

4#2

c· E · |DIF

k ($,!)|2 (2)

where DIFk is the photoelectron dipole matrix element connecting the initial and the final

wave functions:

DIFk = u < "N

I |r|"N!1F ·"el

k >=< !dIF |ru|"el

k > (3)

The photoelectron is described by a plane or Coulomb wave decomposed in spherical

waves basis, |E, l, m >. Each spherical wave is characterized by a certain energy, E =

k2/(2m), and angular momentum, l, m, and contains a radial function, Rkl (spherical Bessel

functions for photodetachment and Coulomb functions for photoionization) and the spherical

harmonics Ylm. The essential quantities that allow one to calculate absolute total and

di$erential (PAD) ionization cross-sections are the photoionization/detachment transition

dipole elements between the Dyson orbital and spherical waves into which the plane wave

of the photoelectron is decomposed.

|DIFk |2 =

""

l,l!=0

l"

m=!l

l!"

m!=!l!

< !dR|ru|E, l, m > · < E, l#, m#|ru|!d

L > ·Ylm(k)Y $l!m!(k) = (4)

1

2#k

""

l,l!=0

l"

m=!l

l!"

m!=!l!

CRkl!m! · (CL

klm)$Ylm(k)Y $l!m!(k), (5)

where !dR and !d

L denote left and right Dyson orbitals, as defined in Ref. [4].

In ezDyson, spherical waves are always computed in the lab frame, which coincides with

the molecular axes frame specified in the input. There are two algorithms for averaging

over molecular orientations: numerical and analytical. In the numerical averaging, the

molecule is rotated with respect to the lab frame, while the lab frame is kept fixed (as

4

defined by the initial orientation of the molecular axes). The code is slower compared to

the analytical averaging, but right now the numerical averaging has been coded with the

additional functions to calculate photoelectron angular distribution. And we used it for

the testing purpose. For the analytical averaging, please see the Appendix for the detailed

derivations.

Once the dipole moment elements are computed, the program calculates absolute to-

tal cross-sections, photoelectron angular distributions (di$erential cross-sections) and %

anisotropy parameters. Total cross-sections are the sum of the square dipole moment ele-

ments multiplied by the Dyson orbitals norm and a prefactor. For isotropic ionization using

non-polarized light, the cross-section is:

" =8#2

3c· E ·

!

d#k · |DIFk |2 (6)

PADs in this program are calculated by integrating over ! in Eqn. (5):

d"

sin$d$=

4#2

c· E ·

1

2#k·"

ll!

min(l,l!)"

m=!min(l,l!)

|CRklm · (CL

kl!m)$| ·%lm($) ·%l!m($) (7)

There is a typo in Eq. (20) in Ref. [5]. The correct expression is Eq. (7) as shown in

the updated manual. In the analytical averaging algorithm, the isotropic cross section is

calculated using:

"(Ek) =4#2&

3ck

"

lmm!

|Ilmm!(k)|2 (8)

Ilmm!(k) =

#

8

3

$1/2 !

dr"d$IF (r)krjl(kr)Ylm(r)Y1m!(r) (9)

Ilmm!(k) =

#

8

3

$1/2 !

dr"d$IF (r)Fl('; kr)Ylm(r)Y1m!(r) (10)

where for the plane wave, jl(kr) is the spherical Bessel function; for the single-center

Coulomb wave, Fl(', kr) is the regular Coulomb function with ' = "1/k, and (l =arg&(l +

1 + i') is the Coulomb phaseshift. The derivation of Eq. (8) is given in Appendix.

5

III. INPUT FILE

A. Generating input files

Input files are in the xml format explained using the example below. The individual

fields can be filled out manually. A Python script is available for the Q-Chem outputs to

be converted to an ezDyson input. However, the details about the photoelectron need to

be specialized. The Python script is located in /xmlsamples and the command format is:

make_xml.py <filename.xml> <Q-Chem_dyson_job.out>

B. Input file structure

We illustrate the input requirements by using the sample file for photodetachment

from F! anion found in /xmlsamples. dyson.xml is the default xml file generated from

fanion.qchem using the Python script, while fanion.xml is the customized (and ready to

run) input file.

----------------------------------------------

<?xml version="1.0" encoding="ISO-8859-1"?>

<root

job = "dyson"

>

----------------------------------------------

The job entry has only one option "dyson" and does not need modification.

---------------------------------------------------

<geometry

n_of_atoms="1"

text = "

F 0.000000 0.000000 0.000000

"

6

/>

---------------------------------------------------

The <geometry> field requires the number of atoms, followed by a list where each line

contains the atom type and the Cartesian coordinates. Only Cartesian coordinates are

supported. If the input is prepared manually, make sure the geometry is given using the

same orientation of the axes as in the electronic structure program that generated Dyson

orbitals.

-------------------------------------------------------------------------------

<free_electron

l_max = "2"

radial_functions="spherical"

radial_functions_possible_valuse="spherical coulomb coulombzerok library"

>

<k_grid n_points="3" min="0.01" max="2.01" />

</free_electron>

-------------------------------------------------------------------------------

The free_electron field refers to the photoelectron description, and it requires the

maximum angular momentum number, l for which spherical waves RklYlm will be generated

and the dipole moment elements calculated. Only functions with l up to 10 (included)

are implemented; however, even this is excesive. For better performance we recommend to

use smaller l_max. As follows from the spherical harmonics expressions, the +m and "m

components should always give identical results. For each l, all possible m projections are

considered. The point group symmetry is not enforced, so if by symmetry certain m values

should lead to zero transition dipoles, check for consistency in the output file. For atoms,

one can save considerable computing time by including only waves with l up to the angular

momentum of the corresponding Dyson orbital plus 1, e.g. l_max = "2" for F!/F.

The radial function should be chosen according to the nature of the ionized core: ”spher-

ical” for neutral core (photodetachment) and ”coulomb” for cationic core (photoionization).

The current implementation of Coulomb waves does not provide good accuracy for low en-

ergy photoelectrons (usually smaller than 0.1 eV), so an asymptotic formula for electron

7

kinetic energies (Ek) approaching zero (threshold) is available by choosing the ”coulombze-

rok” option. This formula is independent of Ek (except for a prefactor: k), so the Ek values

chosen in the next entry are meaningless, but some numbers should be provided for the

program to run properly. The ”library” option is for the debugging purposes only. The

possible_values entries are for guidance and need no modification.

The next line contains the number of Ek’s which should be considered. On the same line

the minimum and maximum Ek should be provided in eV, and the program generates an

equally spaced number of energy points in the range provided. Do not input 0 eV ranges,

or less than two eKe points, the program will crash.

-----------------------------------------------------

<laser

ionization_energy = "3.40" >

<laser_polarisation x="0.0" y="0.0" z="1.0" />

</laser>

-----------------------------------------------------

Next, the laser information depends on the experimental setup represented by the cal-

culation. The ionization energy (in eV) entry is a property of the molecule/anion to be

ionized and is not the total energy of the ionizing radiation. We recommend to use the

best available I.E. (not necessarily corresponding to the theoretical method used to com-

pute Dyson orbitals). The energy of the laser is computed in the program as the sum of the

ionization energy (IE) and each Ek provided in the input. If the absolute cross-sections are

not required (e.g., as in% anisotropy calculations), providing any number is good enough for

running the program.

One should be careful in cross-section calculations when di$erent vibrational transitions

are involved. A recommended option is to provide the I.E. for the 0-0 transition and to

recalculate the cross-sections for other vibrational transitions by rescaling the 0-0 dipole

moments elements.

laser_polarization refers to the polarization of the laser w.r.t. the molecular axes in

the input orientation. In the initial orientation the molecular and lab frame axes coincide,

and if averaging of molecular orientations in lab frame is required the molecule/Dyson or-

bitals are rotated away from the lab frame axes with proper weighting factors included in

8

the final averaged coe!cients.

---------------------------------------------------

<lab_xyz_grid

comment="Axis order: x, y, z">

<axis n_points="173" min="-6.45" max="6.45" />



</lab_xyz_grid>

---------------------------------------------------

lab_xyz_grid data contains information about the xyz Cartesian grid (in A) used for

numerical integration, the calculation of Dyson orbitals and spherical waves. Usually a 0.1

A (sometimes even 0.2 A) spacing on each axis gives good results. Convergence w.r.t. the

density of the xyz grid is quite readily achieved, although one should check for the norm of

the Dyson orbital to make sure the xyz box is large enough. The norm of the Dyson orbitals

from Q-Chem should be 1.0, since Q-Chem prints the renormalized coe!cients in the AO

basis. The real norms of the Dyson orbitals are also printed in the Q-Chem output, and

should be specified in the ezDyson input (as explained below).

-------------------------------------------------------------------

<averaging

method= "analyiso"

method_possible_values="analyiso,noavg, iso, x, y, z, rempi" >

<angles_grid>

<angle comment="alpha" n_points="12" min="0.0" max="2.0" />

<angle comment="beta" n_points="7" min="0.0" max="1.0" />

<angle comment="gamma" n_points="12" min="0.0" max="2.0" />

</angles_grid>

</averaging>

-------------------------------------------------------------------

averaging field needs to be modified. The current default is ”analyiso”, which com-

putes the total cross section for randomly oriented molecules by using Eq. (8). "noavg"

9

computes the dipole matrix elements considering only one orientation of the molecule w.r.t.

laser polarization, as specified in the ezDyson input. "iso" and "x", "y", "z" represent

isotropic averaging and respectively, cylindrical averaging around the x, y, z molecular axes

by a numerical integration. "rempi" is reserved for REMPI (1+n) experiments, where the

molecular orientations with respect to laser polarization have a sin2($) or cos2($) depen-

dence. However, this option has not yet been fully implemented.

The numerical averaging in ("iso", "x", "y", "z") is done by rotating the molecule

w.r.t. the lab frame, by the ()%') angles (in radians/#) provided in the <angles_grid>

entry. Those parameters are only required for numerical everging and will be ignored in

"analyiso" and "noavg". Equally spaced rotations around each of the Euler axes (z, x, z)

are generated within the range provided. Typically, the ranges for isotropic averaging are )

[0-2]#, % [0-1] #, ' [0-2] #. If for any reasons orientations spanning a smaller surface of the

sphere are necessary, modify the angle ranges accordingly. Cyllindrical averaging usually

spans the ) [0-2] # range, and the %, ' angles are not used.

---------------------------------

<job_parameters

unrestricted = "false"

Dyson_MO_transitions = "1"

number_of_MOs_to_plot="0"

MOs_to_plot = ""

/>

---------------------------------

<job_parameters> refers to details of the Dyson calculation by ab initio program e.g.

Q-Chem. For an unrestricted HF calculation, four Dyson orbitals are expected (alpha-

left, beta-left, alpha-right, beta-right), while for the restricted case only two are read (left,

right). Dyson_MO_transitions denotes the order of the ionizing transition that the ezDyson

calculation needs to consider from the list of Dyson orbitals that follows below in the

<dyson_molecular_orbitals> field. If the Python script was used to generate the ezDyson

input, the <dyson_molecular_orbitals> field contains all Dyson orbitals from the Q-Chem

output. Therefore, it is necessary to specify which Dyson orbitals to use for the ezDyson

calculation. For example, assume Dyson orbitals for photodetachment from the O ! ground

10

state to the ground state of O, and to the first two excited states of O are calculated in

Q-Chem. Specifying Dyson_MO_transitions = "2", and assuming unrestricted orbitals,

means that orbitals 5th through 8th are the ones to be used in the ezDyson calculation,

that is the ionization of O! to the first excited state of O.

If number_of_MOs_to_plot is di$erent from 0, a mosplot.dat file is generated, contain-

ing the profile along the z molecular axis of the molecular orbitals specified in MOs_to_plot.

Generally, this feature is useful if one wants to check for oscillatory behavior in the

asymptotic region of the Dyson and/or HF orbitals. The right Dyson orbital profiles

along x, y and z are printed by default in orbital.dat. number_of_MOs_to_plot="0"

and MOs_to_plot = "" are the defaults and no MOs profiles are printed. The for-

mat for generating HF molecular orbitals profiles in mosplot.dat is, for example,

number_of_MOs_to_plot="3" and MOs_to_plot = "5 6 9", for the 5th, 6th and 9th MOs

to be printed.

----------------------------------------------------------

<basis

n_of_basis_functions="52"

purecart="111">

<atom

text = "

F 0

S 6 1.000000

1.14271000E+04 1.80093000E-03

1.72235000E+03 1.37419000E-02

3.95746000E+02 6.81334000E-02

1.15139000E+02 2.33325000E-01

3.36026000E+01 5.89086000E-01

4.91901000E+00 2.99505000E-01

SP 3 1.000000

5.54441000E+01 1.14536000E-01 3.54609000E-02

1.26323000E+01 9.20512000E-01 2.37451000E-01

3.71756000E+00 -3.37804000E-03 8.20458000E-01

11

SP 1 1.000000

1.16545000E+00 1.00000000E+00 1.00000000E+00

SP 1 1.000000

3.21892000E-01 1.00000000E+00 1.00000000E+00

SP 1 1.000000

1.07600000E-01 1.00000000E+00 1.00000000E+00

SP 1 1.000000

3.24096386E-02 1.00000000E+00 1.00000000E+00

SP 1 1.000000

9.76193932E-03 1.00000000E+00 1.00000000E+00

SP 1 1.000000

2.94034317E-03 1.00000000E+00 1.00000000E+00

D 1 1.000000

7.00000000E+00 1.00000000E+00

D 1 1.000000

1.75000000E+00 1.00000000E+00

D 1 1.000000

4.37500000E-01 1.00000000E+00

F 1 1.000000

1.85000000E+00 1.00000000E+00

S 1 1.000000

1.00000000E-14 3.16227770E-11

****

"

/>

</basis>

----------------------------------------------------------

The <basis> field requires the total number of AO basis functions for the molecule

whether pure angular momentum or Cartesian polarization was used (purecart). A 3-4

string of numbers is expected for purecart, specifying if pure (1, e.g. 5d) or Cartesian (2,

e.g. 6d) Gaussians are used for dfgh functions. Default is 1112.

12

Next, a list of the atoms and basis functions in the order of the input in ”geometry”

field is required. The basis set is specified in the Q-Chem format:

X 0

L K scale

)1 CLmin1 CLmin+1

1 ... CLmax1

)2 CLmin2 CLmin+1

2 ... CLmax2

.

.

.

)K CLminK CLmin+1

K ... CLmaxK

****

where

X - Atomic symbol of the atom (atomic number not accepted);

L - Angular momentum symbol (S, P, SP, D, F, G);

K - Degree of contraction of the shell (integer);

scale - Scaling to be applied to exponents (default is 1.00);

)i - Gaussian primitive exponent (positive real number);

CLi - Contraction coe!cient for each angular momentum (non-zero real numbers).

Atoms are terminated with **** No blank lines can be incorporated within the gen-

eral basis set input. Note that more than one contraction coe!cent per line is one required

for compound shells like SP.

----------------------------------------------------------------------------



<dyson_molecular_orbitals>

<DMO norm="0.471723" transition="[REFERENCE|Mu|STATE 1/B1u ]"

comment="dyson left-right alpha"

text="

+2.92856400377020220419E-18

13

-6.17269725710127503737E-17

" />

<DMO norm="0.477153" transition="[REFERENCE|Mu|STATE 1/B1u ]"

comment="dyson right-left alpha"

text="

+4.20626898063226167261E-18

-3.08451802781624482691E-21

" />

</dyson_molecular_orbitals>

----------------------------------------------------------------------------

<dyson_molecular_orbitals> contains a list of the Dyson orbitals coe!cients in AO

basis in the order:

Transition1 left alpha Dyson

Transition1 left beta Dyson (only if ”unrestricted”=1)

Transition1 right alpha Dyson

Transition1 right beta Dyson (only if ”unrestricted”=1)

Transition2 left alpha Dyson

TransitionN right beta Dyson (only if ”unrestricted”=1)

At the top of each Dyson orbital the norm and two comments (transition="" and

comment="") are required. Note: Do not include < or > symbols in the comment lines,

since xml will consider them new input entries. Sometimes Q-Chem outputs contain these

symbols, and one needs to check that they are removed.

-------------------------------------------------------------------------

<molecular_orbitals total_number="52">

<alpha_MOs>

<MO type="alpha" number="1" text="

14

+5.47709686103895809772E-01

" />

...

<MO type="alpha" number="52" text="

-4.09334610487952866773E-16

...

" />

</alpha_MOs>

<beta_MOs>

<MO type="beta" number="1" text="

+5.47709686103895809772E-01

...

" />

...

<MO type="beta" number="52" text="

-4.09334610487952866773E-16

...

" />

</beta_MOs>

</molecular_orbitals>

-------------------------------------------------------------------------

<molecular_orbitals> contains a list of all the alpha and beta molecular orbitals from

the Q-Chem output. This field is not required if no MOs are required to be plotted in

<job_parameters>.

-----------------------------------------------------------------------------



<qchem_input>

--------------------------------------------------------------

15

$molecule

-1 1

F

$end

$rem

$end

$plots

$end

--------------------------------------------------------------

</qchem_input>

</root>

-----------------------------------------------------------------------------

Finally, the Q-Chem input is copied at the end of the xml input for book-keeping pur-

poses, It is not required for ezDyson to run.

IV. RUNNING ezDyson

A. Installation

To extract the files, execute:

tar -xzf ezdyson.v1.tar.gz

A general Linux compiled executable is included (exedys). A few other executables are

available for 32-bit and 64-bit machines, parallel version or not. exedys is the same file

as exedys.32bitpar. Makefiles for di$erent C++ compilers are also provided in the main

directory as makefile. If necessary to recompile, copy the makefile for the proper C++

compiler to makefile and run:

16

make clean

then:

make

B. Running the program

The format for running an ezDyson calculation is:

exedys filename.xml >& filename.out &

In addition to the output file, several other files will be generated depending on the input.

orbital.dat and pad.dat are always generated.

V. OUTPUT

A. Main output

The first part of the output prints out detailed information of the input parameters.

This is useful for checking the input, for example, if the AO basis functions were assigned

properly and the correct columns of Dyson orbitals were read. For computational e!ciency,

the Dyson orbital coe!cients below 1E-06 are treated as zeros.

The first significant output information is the calculated norm of the Dyson orbital

(left*left, right*right and left*right) on the xyz Cartesian grid. One should always check for

convergence of the norm to 1.0 to make sure that the xyz box used for numerical integration

is large enough. In calculations of photoelectron matrix elements, the tail of the Dyson

orbitals plays an important role, because spherical waves have significant density far from

the core.

Xavg Yavg Zavg are the average X, Y and Z coordinates over the left Dyson orbital,

and they define the center of the lab and molecular axes. The center is shifted from the

geometrical center to the center of the left Dyson orbital in order to ensure orthogonality of

the photoelectron wavefunction and Dyson orbital (Ref. [5]). The average square coordinates

<X^2> <Y^2> <Z^2> <R^2> are useful for analyzing the size of the left Dyson orbital along

di$erent directions.

17

The square coe!cients (averaged over molecular orientations with respect to laser polar-

ization, if required in the input) for each of the spherical waves considered are printed in

the order of their wavenumber k (in a.u.) and angular momentum l, m. For each Ek, a sum

of all the square coe!cients is printed next.

Although rarely needed, (averaged) the cross-terms C$klmCkl!m between the waves with

the same angular momentum projection m and di$erent l are also printed in the following

format:

l m l’ cross-term,a.u. l’ cross-term,a.u. ...

Next, the PAD is calculated for ntheta no. of $ angles around the z lab axis, see Eqn. (7).

nkv is the number of photoelectron k values for which the PAD is calculated.

The absolute total cross-section at a certain Ek is just the sum of total coe!cients multi-

plied by the left*right Dyson orbital norms, the ionizing radiation energy (E=IE+Ek) and

a prefactor (valid for isotropic ensembles of molecular orientations and non-polarized light).

The output format is:

E,eV cross-sec1,a.u. cross-sec2,a.u. cross-sec1/cross-sec2

cross-sec1 and cross-sec2 should always be the same. They are calculated by using two

di$erent algorithms for testing purposes. One is by summation of the square coe$cients, the

other by the integration of the di$erential cross-section (PAD) over $ angles, see Eqn. (7).The

second set of printed cross-sections are the absolute cross-section values divided by the

ionizing radiation energy E, and are listed by the energy of the photoelectron Ek in eV. They

are needed for calculations where several vibrational transitions are possible (see Ref. [5]).

% anisotropy parameters are computed using spherical harmonics Y $lm(k)Ylm(k) defined

with the preferential axis along z. If the laser polarization in the input file was di$erent from

0.0 0.0 1.0 [same as the preferential axis for Ylm(k)], the anisotropy results are meaningless.

However, one can use the square and cross coe!cients from ezDyson output and to calculate

separately the PAD and beta anisotropy parameters considering the proper laser polarization

and Ylm(k) frame axis orientations. The columns correspond to:

E,eV sigma_par,a.u. sigma_perp,a.u. sigma_tot,a.u. sigma_tot/cross-sec1

sigma_tot/cross-sec2

18

The last two columns are printed for debugging purposes only. Do not use the sigma_tot

printed here for the total cross-sections. The proper cross-sections are printed above:

cross-sec1 and cross-sec2.

The ezDyson input file is printed at the end of the output.

B. Additional output files

orbital.dat contains the profile of the right alpha Dyson orbital along the three axes

as follows:

x RDys(x,0,0) RDys(x,0,0)*x y RDys(0,y,0) RDys(0,y,0)*y z

RDys(0,0,z) RDys(0,0,z)*z ...

pad.dat contains the PAD calculated using spherical harmonics Ylm(k) with z as the

preferred axis (z laser polarization) in the format:

azimuthal_angle,radians PAD_sq_contrib(lm) PAD_cross_contrib(lml’m)

total_PAD

mosplot.dat is generated only if MO profiles were requested in the input

<job_parameters>. The format is:

z MOi(0,0,z) MOi(0,0,z)*z MOj(0,0,z) MOj(0,0,z)*z ...

For all the above files, all quantities are in atomic units.

VI. EXAMPLES

A. Photoionization

1. H2O

We consider the photoionization of H2O leading to the three lowest states of H2O+.

The initial and final states are described by the CCSD and EOM-IP-CCSD methods. The

following optimized equilibrium geometry is used: rOH = 0.955"

A and % HOH = 104.5o. The

performance using aug-cc-pVTZ and 6-311(3+,3+)G(3df,3pd) basis sets is compared. The

19

total CCSD energies calculated by using aug-cc-pVTZ and 6-311(3+,3+)G(3df,3pd) basis

sets are -76.348752 and -76.353167 hartree, respectively. The cation states were computed

by EOM-IP-CCSD. Table I shows the respective IEs. The corresponding Dyson orbitals

were used to calculate the total cross sections averaging each molecular orientation using

Coulomb wave. Figures 1-3 show the total cross sections corresponding to di$erent electronic

states calculated by using two di$erent basis sets.

TABLE I: IEs (eV) of H2O calculated by EOM-IP-CCSD

Cation state aug-cc-pVTZ 6-311(3+,3+)G(3df,3pd) Experiment

(1b!11 )2B1 12.605 12.598 12.621

(3a!11 )2A1 14.829 14.821

(1b!12 )2B2 18.984 18.994

10 20 30 40 50 60 700.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Photon energy (eV)

Photo

ioniz

atio

n c

ross

sect

ion (

bohr2

)

Synchrotron radiation measurementdipole (e,2e) measurementaug-cc-pVTZ6-311(3+,3+)G(3df,3pd)

(1b1-1)2B

1

FIG. 1: Dyson orbital for the H2O yielding H2O+ (1b!1

1 )2B1 state and the total cross section

compared with the synchrotron radiation and (e,2e) measurements (Ref. [6]).

20

10 20 30 40 50 60 700

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Photon energy (eV)

Photo

ioniz

atio

n c

ross

sect

ion (

bohr2

)

Synchrotron radiation measurement

dipole (e,2e) measurement

aug-cc-pVTZ

6-311(3+,3+)G(3df,3pd)

(3a1-1)2A

1

FIG. 2: Dyson orbital for the H2O yielding H2O+ (3a!1

1 )2A1 state and the total cross section


For H2O+ ionization from the b1 orbital giving rise to the target H2O

+ (1b!11 )2B1 state,

our results agree well with the experimental measurement, except for the resonance region

at low photon energies. The errors of 10-50% are present for the (3a!11 )2A1 and (1b!1

2 )2B2

states. We notice that our model describes the cross sections at high photon energy better

than in the low energy region. The possible reason is that the interaction between the ejected

electron and the core becomes weaker when the free electron carries more photon energy.

In scattering theories, the continuum state wave function is written as the sum of a scatted

wave and a single-center Coulomb wave as a reference state. The present model is missing

the first part. A single charge +1 was placed at the center of frame to represent the electronic

continuum. The scattered wave describes the strong interaction in the core regions, therefore

the correct behavior of this part needs to be obtained from a more advanced model.

21

10 20 30 40 50 60 700

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Photon energy (eV)

Photo

ioniz

atio

n c

ross

sect

ion (

bohr2

)

Synchrotron radiation

dipole (e,2e)

aug-cc-pVTZ

6-311(3+,3+)G(3df,3pd)

(1b2-1)2B

2

FIG. 3: Dyson orbital for the H2O yielding H2O+ (1b!1

2 )2B2 state and the total cross section


2. CO

We consider the photoionization of CO leading to the three lowest states of CO+. The

initial and final states are described by the CCSD and EOM-IP-CCSD methods. The fol-

lowing optimized equilibrium geometry is used: rCO = 1.131"

A. The performance using

aug-cc-pVTZ and 6-311(2+,2+)G(3df,3pd) basis sets is compared. The total CCSD ener-

gies calculated by using aug-cc-pVTZ and 6-311(2+,2+)G(3df,3pd) basis sets are 113.173831

and -113.184215 hartree, respectively. The cation states were computed by EOM-IP-CCSD.

Table II shows the respective IEs. Figures 4-6 show the total cross sections corresponding

to di$erent electronic states calculated by using either the Coulomb wave or plane wave to

describe the final continuum state.

The calculated cross section using Coulomb wave does not agree with the experimental

measurement for the (5")!1X2'+ state. Plane wave calculation only shows the correct

22

TABLE II: IEs (eV) of CO calculated by EOM-IP-CCSD


(5!)!1X2!+ 14.194 14.180 14.014

(1")!1A2" 17.066 17.057

(4!)!1B2! 19.796 19.775

10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

Photon energy (eV)

Photo

ioniz

atio

n c

ross

sect

ion (

bohr2

)

Sychrotron radiationaug-cc-pVTZ / Coulomb6-311(2+,2+)G(3df,3pd) / Coulombaug-cc-pVTZ / plane6-311(2+,2+)G(3df,3pd) / plane

(5!)-1X2"

+

FIG. 4: Dyson orbital for the CO yielding the (5!)!1X2!+ state and the total cross section by

using either Coulomb wave or plane wave to describe the final continuum state. The results using

two di#erent basis sets are compared with the synchrotron radiation experiment (Ref. [10]).

behavior of cross sections at the high photon energy region. The errors of 10-50% are

observed for (1#)!1A2( and (4")!1B2' states using the Coulomb waves.

23

15 20 25 30 35 40 45 500.1

0.2

0.3

0.4

0.5

0.6

Photon energy (eV)

Photo

ioniz

atio

n c

ross

sect

ion (

bohr2

)

Synchrotron radiationaug-cc-pVTZ / Coulomb6-311(2+,2+)G(3df,3pd) / Coulombaug-cc-pVTZ / plane6-311(2+,2+)G(3df,3pd) / plane

(1#)-1A2$

FIG. 5: Dyson orbital for the CO yielding the (1")!1A2" state and the total cross section by using

either Coulomb wave or plane wave to describe the final continuum state. The results using two

di#erent basis sets are compared with the synchrotron radiation experiment (Ref. [10]).

3. CH4

We consider the photoionization of CH4 leading to the two lowest states of CH+4 . The

initial and final states are described by the CCSD and EOM-IP-CCSD methods. The follow-

ing optimized equilibrium geometry is used: rCH = 1.087"

A. The performance using aug-

cc-pVTZ and 6-311(3+,3+)G(3df,3pd) basis sets is compared. The total CCSD energies

calculated by using aug-cc-pVTZ and 6-311(3+,3+)G(3df,3pd) basis sets are -113.173831

and -40.454779 hartree, respectively. The cation states were computed by EOM-IP-CCSD.

Table III shows the respective IEs. Here we used the experimental I.E. to calculate the total

cross sections. Figure 7 shows the total cross section corresponding to the lowest B1/B2/B3

electronic state by using either Coulomb wave or plane wave to describe the final continuum

state.

24

15 20 25 30 35 40 45 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Photon energy (eV)

Photo

ioniz

atio

n c

ross

sect

ion (

bohr2

)

Synchrotron radiation

aug-cc-pVTZ / Coulomb

6-311(2+,2+)G(3df,3pd) / Coulomb

aug-cc-pVTZ / plane

6-311(2+,2+)G(3df,3pd) / plane

(4!)-1B2"

FIG. 6: Dyson orbital for the CO yielding the (4!)!1B2! state and the total cross section by using

either Coulomb wave or plane wave to describe the final continuum state. The results using two

di#erent basis sets are compared with the synchrotron radiation experiment (Ref. [10]).

TABLE III: IEs (eV) of CH4 calculated by EOM-IP-CCSD


A 23.381 23.371

B1/B2/B3 14.403 14.402 12.61

The calculated cross section using Coulomb wave does not rise from zero as increasing

the photon energy. The plane wave result shows the correct behavior in the low photon

energy region and closer to the experimental data. It is because Coulomb wave has the

wrong boundary condition representing the final continuum of CH4 photoionization.

25

12 13 14 15 16 17 180

0.2

0.4

0.6

0.8

1

Photon energy (eV)

Photo

ioniz

atio

n c

ross

sect

ion (

bohr2

)

experiment


aug-cc-pVTZ / plane

6-311(3+,3+)G(3df,3pd) / Coulomb

6-311(3+,3+)G(3df,3pd) / plane

CH4

FIG. 7: Total cross section for CH4 yielding the lowest CH+4 B1/B2/B3 state by using either

Coulomb wave or plane wave to describe the final continuum state. The results using two di#erent

basis sets are compared with the experimental measurements (Ref. [9]).

4. CH3OH

We consider the photoionization of CH3OH leading to the six lowest states of CH3OH+.


performance using the aug-cc-pVTZ and the 6-311(3+,3+)G(3df,3pd) basis sets is compared.

The total CCSD energies calculated by using aug-cc-pVTZ and 6-311(3+,3+)G(3df,3pd)

basis sets are -115.580797 and -115.586578 hartree respectively. The cation states were com-

puted by EOM-IP-CCSD. Table IV shows the respective IEs. Here we used the experimental

I.E. to calculate the total cross sections. Figure 8 shows the total cross section corresponding

to the lowest A## electronic state by using either Coulomb wave or plane wave to describe

the final continuum state.

The CH3OH photoionization cross sections show the same phenomena as the CH4 pho-

toionization because of the wrong boundary condition applied by using Coulomb wave. The

result using plane wave rises slower than the experimental data as increasing the photon

energy. Overall, the absolute values are within the range of experimental data.

26

TABLE IV: IEs (eV) of CH3OH calculated by EOM-IP-CCSD


1A# 12.798 12.793

2A# 15.306 15.301

3A# 17.758 17.757

4A# 23.136

1A## 11.010 11.001 10.84

2A## 15.829 15.826

10 10.5 11 11.5 120

0.1

0.2

0.3

0.4

0.5

Photon energy (eV)

Photo

ioniz

atio

n c

ross

sect

ion (

bohr2

)

experiment


aug-cc-pVTZ / plane

6-311(3+,3+)G(3df,3pd) / Coulomb

6-311(3+,3+)G(3df,3pd) / plane

CH3OH

FIG. 8: Total cross section for CH3OH yielding the lowest CH3OH+ A## state by using either

Coulomb wave or plane wave to describe the final continuum state. The results using two di#erent


5. CH2O

We consider the photoionization of CH2O leading to the five lowest states of CH2O+.


performance using aug-cc-pVTZ and 6-311(3+,3+)G(3df,3pd) basis sets is compared. The

total CCSD energies calculated by using aug-cc-pVTZ and 6-311(3+,3+)G(3df,3pd) basis

sets are -114.358081 and -114.365504 hartree respectively. The cation states were computed

27

by EOM-IP-CCSD. Table V shows the respective IEs. Figures 9-10 shows the total cross

section corresponding to the four lowest electronic states by using either Coulomb wave or

plane wave to describe the final continuum state.

TABLE V: IEs (eV) of CH2O calculated by EOM-IP-CCSD


1A1 16.159 16.146

2A1 21.800 21.783

1B1 10.854 10.842 10.88

2B1 17.554 17.550

1B2 14.699 14.692

10 12 14 16 18 20 22 240

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Photon energy (eV)

Photo

ioniz

atio

n c

ross

sect

ion (

bohr2

)

experiment

1B1

1B2

1A1

2B1

1B1+1B2

1B1

1B2

1A1

2B1

1B1+1B2

solid line: 6-311(3+,3+)G(3df,3pd)dash line: aug-cc-pVTZ

CH2O / Coulomb

FIG. 9: Total cross section for CH2O yielding the four lowest CH2O+ 1B1, 1B2, 1A1 and 2B1 states

by using Coulomb wave to describe the final continuum state. The results using two di#erent basis

sets are compared with the experimental measurements (Ref. [9]).

In the range of the photon energy used in experiments, the photoionization of CH2O

yields four CH2O+ states. For Coulomb wave, the calculated cross section does not rise

from zero, and the sum of cross sections from1B1 and 1B2 states is close to the experimental

data. For plane wave, the sum of cross sections from 1B1 and 1B2 states overestimates the

experimental results by 80%.

28

10 12 14 16 18 20 22 240

0.2

0.4

0.6

0.8

1

1.2

1.4

Photon energy (eV)

Photo

ioniz

atio

n c

ross

sect

ion (

bohr2

)

experiment1B11B21A12B11B1+1B21B11B21A12B11B1+1B2

solid line: 6-311(3+,3+)G(3df,3pd)dash line: aug-cc-pVTZ

CH2O / Plane

FIG. 10: Total cross section for CH2O yielding the four lowest CH2O+ 1B1, 1B2, 1A1 and 2B1

states by using plane wave to describe the final continuum state. The results using two di#erent


6. C2H4 photoionization

We consider the photoionization of C2H4 leading to the six lowest states of C2H4+. The

initial and final states are described by the CCSD and EOM-IP-CCSD methods. The perfor-

mance using aug-cc-pVTZ and 6-311(3+,3+)G(3df,3pd) basis sets is compared. The total

CCSD energies calculated by using aug-cc-pVTZ and 6-311(3+,3+)G(3df,3pd) basis sets

are -78.464926 and -78.470005 hartree respectively. The cation states were computed by

EOM-IP-CCSD. Table VI shows the respective IEs. Here we used the experimental I.E. to

calculate the total cross sections. Figure 11 shows the total cross section corresponding to

the lowest B1u electronic state by using either Coulomb wave or plane wave to describe the

final continuum state.

The experimental data are in the low photon energy region. The calculated cross section

using plane wave rises as increasing the photon energy. But it does not start from zero

as shown in the experiments. Both plane wave and Coulomb wave fail to reproduce the

experimental results in the low photon energy region.

29

TABLE VI: IEs (eV) of C2H4 calculated by EOM-IP-CCSD


1Ag 14.945 14.946

2Ag 24.468

B1g 13.163 13.169

B1u 10.709 10.700 10.99

B2u 16.351 16.352

B3u 19.648 19.643

10.8 11 11.2 11.4 11.6 11.8 12 12.2 12.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Photon energy (eV)

Photo

ioniz

atio

n c

ross

sect

ion (

bohr2

)

experiment


aug-cc-pVTZ / plane

6-311(3+,3+)G(3df,3pd) / Coulomb

6-311(3+,3+)G(3df,3pd) / plane

C2H

4

FIG. 11: Total cross section for C2H4 yielding the lowest C2H4+ B1u state by using either Coulomb

wave or plane wave to describe the final continuum state. The results using two di#erent basis sets

are compared with the experimental measurements (Ref. [9]).

B. Photodetachment

We used the updated code (analytical averaging plus some minor bugs fixed) to repeat a

few calculations in Ref. [5] and no di$erences are observed.

30

1. H!

For the photodetachment of H!, the initial and final states are described by the CCSD

and EOM-IP-CCSD methods. The total CCSD energies of H! anion calculated using aug-cc-

pVDZ, aug-cc-pVTZ and 6-311(3+)G(3pd) basis sets are -0.524029, -0.526562 and -0.52569

hartree, and the threshold energy values are 0.672, 0.728 and 0.704 eV, respectively. Fig-

ure 12 shows the calculated cross sections using di$erent basis sets compared with the

experimental measurements.

0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Photon energy (eV)

Photo

deta

chm

ent cr

oss

sect

ion (

bohr2

)

Exact ECS calculation and experiment

6-311(.,3+)G(3pd)

aug-cc-pVTZ

aug-cc-pVDZ

H-

FIG. 12: Total cross section for photodetachment of H! calculated using di#erent basis sets. The

results are compared with the exact Exterior Complex Scaling (ECS) calculations (Ref. [8]) and

experimental measurements (Ref. [12]).

2. Li!

For the photodetachment of Li!, the initial and final states are described by the CCSD

and EOM-IP-CCSD methods. The total CCSD energy of Li! anion calculated using 6-

311(3+)G(3df) basis sets is -7.469909 hartree, and the threshold energy value is 0.618 eV.

Figure 13 shows the calculated cross sections compared with experimental measurements.

31

0.5 1 1.5 2 2.50

1

2

3

4

5

6

Photon energy (eV)

Photo

deta

chm

ent cr

oss

sect

ion (

bohr2

)

experiment6-311(3+,.)G(3df,.)

Li-

FIG. 13: Total cross section for photodetachment of Li! compared with experimental measurements

(Ref. [7]).

3. C!

For the photodetachment of C!, the initial and final states are described by the CCSD

and EOM-IP-CCSD methods. The total CCSD energy of C! anion calculated using 6-

311(3+)G(3df) basis sets is -37.839098 hartree, and the threshold energy value is 1.329

eV. Figure 14 shows the calculated cross sections using di$erent basis sets compared with

experimental measurements.

Appendix A: Electric dipole interaction operator in laboratory frame

The light source and the electron detector are in the laboratory frame. If the linear

polarized light is used, the polarization direction is assumed to be along the z axis (mp = 0).

For the left (mp = 1) and right (mp = "1) circular polarized light, the polarizations are

defined in xy plane. The set of Euler angles R = {), %, '} carries the molecular frame into

coincidence with laboratory frame. To average over molecular orientations with respect to

three Euler angles, we need to write the electric dipole interaction operator in the laboratory

frame through the following procedures:

An irreducible tensor with the rank k is an operator of 2k + 1 components Tkq which

32

1.5 2 2.5 3 3.50.1

0.2

0.3

0.4

0.5

0.6

Photon energy (eV)

Photo

deta

chm

ent cr

oss

sect

ion (

bohr2

)

experiment

6-311(3+,.)G(3df,.)

C-

FIG. 14: Total cross section for photodetachment of C! compared with experimental measurements

(Ref. [11]).

tansforms under a rotation {), %, '} of axes:

DTkqD+ =

"

p

TkqDkpq()%') (A1)

where D represents an infinitesimal rotation (1"i)J!) using angular momentum operator

J!. Thus, the matrix elements of D are:

Dkpq = #kp| 1 " i)J! |kq$ = *pq " i) #kp| J! |kq$ (A2)

Therefore, Eq. (A1) becomes:

J!Tkq " TkqJ! ="

p

Tkq #kp| J! |kq$ (A3)

Taking + = z or ± and using the matrix elements of Jz and J±, we obtain the following

commutation rules:

[Jz, Tkq] = qTkq

[J±, Tkq] = [(k ± q + 1)(k % q)]1/2Tkq±1, (A4)

which can be used to construct a tensor’s spherical components from its Cartesian com-

ponents. For example, vector A can be treated as a spherical tensor of rank k = 1 with

33

q = 0,±1. The three Cartesian components of A satisfy the commutation relations with

angular momentum Jz:

[Jz, Az] = 0

[Jx, Az] = "iAy

[Jy, Az] = iAx (A5)

Let q = 0,±1 in Eq. (A4) combined with Eq. (A5). This yields:

A0 = Az

A±1 =1!2[J±, A0]

=1!2

([Jx, Az] ± i[Jy, Az])

= %1!2

(Ax ± iAy) (A6)

Therefore, we can write the dipole operator r as a spherical tensor defined in the molecular

frame:

r10 = z =

%

4#

3rY 0

1

r1±1 = %1!2

(x ± iy) =

%

4#

3rY ±1

1 (A7)

and light polarization vector as a spherical tensor defined in the laboratory frame:

,10 = ,z

,1±1 = %1!2

(,x ± i,y) (A8)

The transformation of a spherical tensor components from the molecular frame into the

laboratory frame can be obtained through the Euler rotational matrix:

r#1m! ="

m

r1mD1mm!(R) (A9)

Because the left circular polarization mp = 1 corresponds to 1&2(,x + i,y), right circular

polarization mp = "1 corresponds to 1&2(,x " i,y) and the linear polarization mp = 0

corresponds to ,z, the electric dipole interaction operator in the laboratory frame is:

-mp(r, R) = r# · =

#

4#

3

$1/2

(1 " 2*1mp)r"

m!

Y1m!(r)D1m!mp

(R") (A10)

34

Finally, with the consideration of di$erent molecular orientations, the cross section can

be written as:

d" =4#2&

ck

&

&

&I(k, R)

&

&

&

2dEkdkdR (A11)

where k is the outgoing electron’s momentum measured in the molecular frame

along the direction {$,!}, and I(k, R) is the photoelectron matrix element I(k, R) =

#"dIF (r)

&

&-mp(r, R)&

&"(!)(r,k)$.

Appendix B: Total cross sections for isotropic averaging molecular orientations

To derive Eq.(8), we combine Eq. (A10), (A11), and use the following angular momentum

coupling relations:

Y $l!m!(k)Ylm(k) = ("1)m!

'

(2l# + 1)(2l + 1)

4#

(1/2"

K

)

*

+

(lm, l# " m#|KM)(l0, l#0|K0)

(2K + 1)!1/2YKM(k)

,

-

.

(B1)

D1$m!

!mp(R)D1

m!mp(R) = ("1)m!

!!mp

"

K!

(1m", 1 " m#"|K"M")(1mp, 1 " mp|K"0)DK!

M!0(R)

(B2)

The doubly di$erential cross section at the ejected electron’s energy Ek, with respect to

the electron angular distribution in the molecular frame and the molecular orientation in

the laboratory frame, is written as a double sum of the matrix element Ilmm!(k) computed

in the molecular frame:

d"(Ek)

dkdR=

4#2&

ck

"

lmm!

"

l!m!m!!

)

/

/

/

/

/

*

/

/

/

/

/

+

("1)m!+m!

!!mp

0

(2l!+1)(2l+1)4#

11/2il

!!lei($l!$l! )I$l!m!m!

!Ilmm!

2

K(lm, l# " m#|KM)(l0, l#0|K0)(2K + 1)!1/2YKM(k)

2

K!

(1m", 1 " m#"|K"M")(1mp, 1 " mp|K"0)DK!

M!0(R)

,

/

/

/

/

/

-

/

/

/

/

/

.

(B3)

where for the plane wave,

Ilmm!(k) =

#

8

3

$1/2 !

dr"d$IF (r)krjl(kr)Ylm(r)Y1m!(r) (B4)

and for the Coulomb wave,

35

Ilmm!(k) =

#

8

3

$1/2 !

dr"d$IF (r)Fl('; kr)Ylm(r)Y1m!(r) (B5)

The symbol (l1m1, l2m2|l3m3) denotes the Clebsch"Gordan coe!cient that satisfies the

triangular relations m3 = m1 + m2 and |l1 " l2| & l3 & |l1 + l2|. Therefore, K varies from

0 to 2lmax and K" varies from 0 to 2. The averaging over three Euler angles of R and

integration over two polar angles of k are:

1

8#2

!

dRDK!

M!0(R) = *K!0*M!0 (B6)

!

dkYKM(k) = (4#)1/2 *K0*M0 (B7)

which gives K", M", K and M = 0, and the Clebsch"Gordan coe!cient at these condi-

tions is:

(lm, l " m|00) = ("1)l!m(2l + 1)!1/2 (B8)

Therefore, all the cross terms vanish and the total cross section for the random target

orientation is:

"(Ek) =4#2&

3ck

"

lmm!

|Ilmm!(k)|2 (B9)

[1] B.T. Pickup and O. Goscinsky, Mol. Phys. 26, 1013 (1973).

[2] B.T. Pickup, Chem. Phys. 19, 193 (1977).

[3] Y. Shao, L.F. Molnar, Y. Jung, J. Kussmann, C. Ochsenfeld, S. Brown, A.T.B. Gilbert, L.V.

Slipchenko, S.V. Levchenko, D.P. O’Neil, R.A. Distasio Jr, R.C. Lochan, T. Wang, G.J.O.

Beran, N.A. Besley, J.M. Herbert, C.Y. Lin, T. Van Voorhis, S.H. Chien, A. Sodt, R.P. Steele,

V.A. Rassolov, P. Maslen, P.P. Korambath, R.D. Adamson, B. Austin, J. Baker, E.F.C. Bird,

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