.. index::
single: Linear molecules; Supersymmetry
single: Symmetry
.. _TUT\:sec\:x2:
Computing high symmetry molecules
=================================
.. only:: html
.. contents::
:local:
:backlinks: none
|molcas| makes intensive use of the symmetry properties of the
molecular systems in all parts of the calculation. The highest
symmetry point group available,
however, is the |Dth| point group, which makes things somewhat
more complicated when the molecule has higher symmetry.
One of such cases is the calculation of linear molecules.
In this section we describe calculations on
different electronic states of three diatomic molecules:
:math:`\ce{NiH}`, a heteronuclear molecule which belongs to the |Cinfv|
symmetry group and :math:`\ce{C2}` and :math:`\ce{Ni2}`, two homonuclear molecules which belong
to the |Dinfh| symmetry group. They must be computed in
|molcas| using the lower order symmetry groups |Ctv| and
|Dth|, respectively, and therefore some codes such RASSCF use specific tools
to constrain the resulting wave functions
to have the higher symmetry of the actual point group.
It must be pointed out clearly that linear symmetry cannot always be fully
obtained in |molcas| because the tools to average over degenerate representations
are not totally implemented presently in the
RASSCF program. This is the case, for instance, for the :math:`\delta`
orbitals in a |Ctv|\--\ |Cinfv| situation, as will be shown below.
(For problems related to accurate calculations of diatomic
molecules and symmetry see Ref. :cite:`Partridge:95` and
:cite:`Taylor:92a`, respectively.).
In a final section we will briefly comment the situation of
high symmetry systems other than linear.
.. index::
single: Linear molecules; NiH
single: NiH
.. _TUT\:sec\:nih:
A diatomic heteronuclear molecule: :math:`\ce{NiH}`
---------------------------------------------------
Chemical bonds involving transition-metal atoms are often
complex in nature due to the common presence of several unpaired
electrons resulting in many close-lying spectroscopic states
and a number of different factors such spin--orbit coupling or the
importance of relativistic effects. :math:`\ce{NiH}` was the first
system containing a transition-metal atom to be studied with
the CASSCF method :cite:`Roos:82`. The large dynamic correlation
effects inherent in a 3d semi-occupied shell with many electrons is
a most severe problem, which few methods have been able to compute. The
calculated dipole moment of the system has become one measurement
of the quality of many *ab initio* methods :cite:`Roos:87`.
We are not going to analyze the effects in detail. Let us only
say that an accurate treatment of the correlation effects
requires high quality methods such as MRCI, ACPF or CASPT2,
large basis sets, and an appropriate treatment of
relativistic effects, basis set superposition errors,
and core-valence correlation. A detailed CASPT2 calculation
of the ground state of :math:`\ce{NiH}` can be found elsewhere :cite:`Pou:94`.
The :math:`^3F` (3d\ :math:`^8`\4s\ :math:`^2`) and :math:`^3D` (3d\ :math:`^9`\4s\ :math:`^1`) states of the nickel
atom are almost degenerate with a splitting of only 0.03 eV :cite:`Andersson:92c` and
are characterized by quite different chemical behavior. In systems such as the
:math:`^2\Delta` ground state of :math:`\ce{NiH}` molecule, where both states take part in the
bonding, an accurate description of the low-lying :math:`\ce{Ni}` atomic states is required.
The selection of the active space for :math:`\ce{NiH}` is not trivial.
The smallest set of active orbitals for the :math:`^2\Delta` ground state which allows
a proper dissociation and also takes into account the important 3d\ :math:`\sigma`
correlation comprises the singly occupied 3\ |dxy| orbital and three :math:`\sigma`
orbitals (3\ |dzt|, :math:`\sigma`, and :math:`\sigma^*`). One cannot however expect to obtain
accurate enough molecular properties just by including non-dynamical correlation
effects. MRCI+Q calculations with the most important CASSCF configurations
in the reference space proved that at least one additional 3d\ :math:`\delta` (3\ |dxtyt|)
and its correlating orbital were necessary to obtain spectroscopic constants in close
agreement with the experimental values. It is, however, a larger active space comprising
all the eleven valence electrons distributed in twelve active orbitals
(:math:`\sigma`, :math:`\sigma^*`, d, d') that is the most consistent choice of active orbitals as
evidenced in the calculation of other metal hydrides such as :math:`\ce{CuH}` :cite:`Pou:94`
and in the electronic spectrum of the :math:`\ce{Ni}` atom :cite:`Andersson:92c`. This is the
active space we are going to use in the following example. We will use the
ANO-type basis set contracted to :math:`\ce{Ni}` [5s4p3d1f] / :math:`\ce{H}` [3s2p] for simplicity. In
actual calculations g functions on the transition metal and d functions on
the hydrogen atom are required to
obtain accurate results.
.. index::
single: Spherical Harmonics; C∞v
First we need to know the behavior of each one of the basis functions
within each one of the symmetries. Considering the molecule placed in
the :math:`z` axis the classification of the spherical harmonics into the |Cinfv|
point group is:
.. table:: Classification of the spherical harmonics in the |Cinfv| group.
:name: tab:cinfv
============== ======= ======= ======= ======= ======= =======
Symmetry Spherical harmonics
============== ===============================================
:math:`\sigma` |s| |pz| |dzt| |fztt|
:math:`\pi` |px| |py| |dxz| |dyz| |fx| |fy|
:math:`\delta` |dxtyt| |dxy| |fxyz| |fz|
:math:`\phi` |fxtt| |fytt|
============== ======= ======= ======= ======= ======= =======
.. index::
single: Diatomic molecules; Symmetry problems
single: Spherical Harmonics; C2v
In |Ctv|, however, the functions are distributed into the four representations
of the group and therefore different symmetry representations can be mixed.
The next table lists the distribution of the
functions in |Ctv| and the symmetry of the corresponding orbitals in |Cinfv|.
.. table:: Classification of the spherical harmonics and |Cinfv| orbitals in the |Ctv| group.
:name: tab:c2v
============ ======================== ======================== ======================== ======================== ======================== ========================
Symm.\ [#a]_ Spherical harmonics (orbitals in |Cinfv|)
============ =====================================================================================================================================================
|ao| (1) |s| (:math:`\sigma`) |pz| (:math:`\sigma`) |dzt| (:math:`\sigma`) |dxtyt| (:math:`\delta`) |fztt| (:math:`\sigma`) |fz| (:math:`\delta`)
|bo| (2) |px| (:math:`\pi`) |dxz| (:math:`\pi`) |fx| (:math:`\pi`) |fxtt| (:math:`\phi`)
|bt| (3) |py| (:math:`\pi`) |dyz| (:math:`\pi`) |fy| (:math:`\pi`) |fytt| (:math:`\phi`)
|at| (4) |dxy| (:math:`\delta`) |fxyz| (:math:`\delta`)
============ ======================== ======================== ======================== ======================== ======================== ========================
.. [#a] In parenthesis the number of the symmetry in |molcas|. It depends on the generators used in :program:`SEWARD`.
In symmetry |ao| we find both :math:`\sigma` and :math:`\delta` orbitals. When the
calculation is performed in |Ctv| symmetry all the orbitals of |ao| symmetry
can mix because they belong to the same representation, but this is not
correct for |Cinfv|. The total symmetry must be kept |Cinfv| and therefore the
:math:`\delta` orbitals should not be allowed to rotate and mix with the :math:`\sigma`
orbitals. The same is true in the |bo| and |bt| symmetries with the :math:`\pi` and
:math:`\phi` orbitals, while in |at| symmetry this problem does not exist because
it has only :math:`\delta` orbitals (with a basis set up to f functions).
The tool to restrict possible orbital rotations is the option :kword:`SUPSym` in the
RASSCF program. It is important to start with clean orbitals belonging to the
actual symmetry, that is, without unwanted mixing.
But the problems with the symmetry are not solved with the :kword:`SUPSym` option only.
Orbitals belonging to different components of a degenerate representation should also be
equivalent. For example: the :math:`\pi` orbitals in |bo| and |bt| symmetries should have the
same shape, and the same is true for the :math:`\delta` orbitals in |ao| and |at| symmetries.
This can only be partly achieved in the RASSCF code. The input option :kword:`AVERage`
will average the density matrices for representations |bo| and |bt| (:math:`\pi` and :math:`\phi`
orbitals), thus producing equivalent orbitals. The present version does not, however,
average the :math:`\delta` orbital densities in representations |ao| and |at| (note that
this problem does not occur for electronic states with an equal occupation of the
two components of a degenerate set, for example :math:`\Sigma` states).
A safe way to obtain totally symmetric orbitals is to reduce the symmetry to :math:`C_1`
(or :math:`C_s` in the homonuclear case) and perform a state-average calculation for the
degenerate components.
.. index::
single: Spherical Harmonics; MOLCAS format
We need an equivalence table to know the correspondence of
the symbols for the functions in |molcas| to the spherical harmonics (SH):
.. table:: |molcas| labeling of the spherical harmonics.
:name: tab:labels
======== ======= ======== ======= ======== =======
|molcas| SH |molcas| SH |molcas| SH
======== ======= ======== ======= ======== =======
1s |s| 3d2+ |dxtyt| 4f3+ |fxtt|
2px |px| 3d1+ |dxz| 4f2+ |fz|
2pz |pz| 3d0 |dzt| 4f1+ |fx|
2py |py| 3d1\ |-| |dyz| 4f0 |fztt|
|zws| 3d2\ |-| |dxy| 4f1\ |-| |fy|
|zws| 4f2\ |-| |fxyz|
|zws| 4f3\ |-| |fytt|
======== ======= ======== ======= ======== =======
We begin by performing a SCF calculation and analyzing the resulting
orbitals. The employed bond distance is close
to the experimental equilibrium bond length for the ground state :cite:`Pou:94`.
Observe in the following SEWARD input that the symmetry generators,
planes :math:`yz` and :math:`xz`, lead to a |Ctv| representation. In the SCF
input we have used the option :kword:`OCCNumbers` which allows specification of
occupation numbers other than 0 or 2. It is still the closed shell SCF
energy functional which is optimized, so the obtained SCF energy has no
physical meaning. However, the computed orbitals are somewhat
better for open shell cases as :math:`\ce{NiH}`. The energy of the virtual orbitals
is set to zero due to the use of the :kword:`IVO` option.
The order of the orbitals may change in different computers
and versions of the code.
.. index::
single: Program; Seward
single: Program; SCF
single: SCF; OccNumbers
single: IVO
.. extractfile:: advanced/SCF.NiH.input
&SEWARD
Title
NiH G.S
Symmetry
X Y
Basis set
Ni.ANO-L...5s4p3d1f.
Ni 0.00000 0.00000 0.000000 Bohr
End of basis
Basis set
H.ANO-L...3s2p.
H 0.000000 0.000000 2.747000 Bohr
End of basis
End of Input
&SCF
TITLE
NiH G.S.
OCCUPIED
8 3 3 1
OCCNumber
2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0
2.0 2.0 2.0
2.0 2.0 2.0
1.0
::
SCF orbitals + arbitrary occupations
Molecular orbitals for symmetry species 1
ORBITAL 4 5 6 7 8 9 10
ENERGY -4.7208 -3.1159 -.5513 -.4963 -.3305 .0000 .0000
OCC. NO. 2.0000 2.0000 2.0000 2.0000 2.0000 .0000 .0000
1 NI 1s0 .0000 .0001 .0000 -.0009 .0019 .0112 .0000
2 NI 1s0 .0002 .0006 .0000 -.0062 .0142 .0787 .0000
3 NI 1s0 1.0005 -.0062 .0000 -.0326 .0758 .3565 .0000
4 NI 1s0 .0053 .0098 .0000 .0531 -.4826 .7796 .0000
5 NI 1s0 -.0043 -.0032 .0000 .0063 -.0102 -.0774 .0000
6 NI 2pz .0001 .0003 .0000 -.0015 .0029 .0113 .0000
7 NI 2pz -.0091 -.9974 .0000 -.0304 .0622 .1772 .0000
8 NI 2pz .0006 .0013 .0000 .0658 -.1219 .6544 .0000
9 NI 2pz .0016 .0060 .0000 .0077 -.0127 -.0646 .0000
10 NI 3d0 -.0034 .0089 .0000 .8730 .4270 .0838 .0000
11 NI 3d0 .0020 .0015 .0000 .0068 .0029 .8763 .0000
12 NI 3d0 .0002 .0003 .0000 -.0118 -.0029 -.7112 .0000
13 NI 3d2+ .0000 .0000 -.9986 .0000 .0000 .0000 .0175
14 NI 3d2+ .0000 .0000 .0482 .0000 .0000 .0000 .6872
15 NI 3d2+ .0000 .0000 .0215 .0000 .0000 .0000 -.7262
16 NI 4f0 .0002 .0050 .0000 -.0009 -.0061 .0988 .0000
17 NI 4f2+ .0000 .0000 .0047 .0000 .0000 .0000 -.0033
18 H 1s0 -.0012 -.0166 .0000 .3084 -.5437 -.9659 .0000
19 H 1s0 -.0008 -.0010 .0000 -.0284 -.0452 -.4191 .0000
20 H 1s0 .0014 .0007 .0000 .0057 .0208 .1416 .0000
21 H 2pz .0001 .0050 .0000 -.0140 .0007 .5432 .0000
22 H 2pz .0008 -.0006 .0000 .0060 -.0093 .2232 .0000
ORBITAL 11 12 13 14 15 16 18
ENERGY .0000 .0000 .0000 .0000 .0000 .0000 .0000
OCC. NO. .0000 .0000 .0000 .0000 .0000 .0000 .0000
1 NI 1s0 -.0117 -.0118 .0000 .0025 .0218 -.0294 .0000
2 NI 1s0 -.0826 -.0839 .0000 .0178 .1557 -.2087 .0000
3 NI 1s0 -.3696 -.3949 .0000 .0852 .7386 -.9544 .0000
4 NI 1s0 -1.3543 -1.1537 .0000 .3672 2.3913 -2.8883 .0000
5 NI 1s0 -.3125 .0849 .0000 -1.0844 .3670 -.0378 .0000
6 NI 2pz -.0097 -.0149 .0000 .0064 .0261 -.0296 .0000
7 NI 2pz -.1561 -.2525 .0000 .1176 .4515 -.4807 .0000
8 NI 2pz -.3655 -1.0681 .0000 .0096 1.7262 -2.9773 .0000
9 NI 2pz -1.1434 -.0140 .0000 -.1206 .2437 -.9573 .0000
10 NI 3d0 -.1209 -.2591 .0000 .2015 .5359 -.4113 .0000
11 NI 3d0 -.3992 -.3952 .0000 .1001 .3984 -.9939 .0000
12 NI 3d0 -.1546 -.1587 .0000 -.1676 -.2422 -.4852 .0000
13 NI 3d2+ .0000 .0000 -.0048 .0000 .0000 .0000 -.0498
14 NI 3d2+ .0000 .0000 -.0017 .0000 .0000 .0000 -.7248
15 NI 3d2+ .0000 .0000 .0028 .0000 .0000 .0000 -.6871
16 NI 4f0 -.1778 -1.0717 .0000 -.0233 .0928 -.0488 .0000
17 NI 4f2+ .0000 .0000 -1.0000 .0000 .0000 .0000 -.0005
18 H 1s0 1.2967 1.5873 .0000 -.3780 -2.7359 3.8753 .0000
19 H 1s0 1.0032 .4861 .0000 .3969 -.9097 1.8227 .0000
20 H 1s0 -.2224 -.2621 .0000 .1872 .0884 -.7173 .0000
21 H 2pz -.1164 -.4850 .0000 .3388 1.1689 -.4519 .0000
22 H 2pz -.1668 -.0359 .0000 .0047 .0925 -.3628 .0000
Molecular orbitals for symmetry species 2
ORBITAL 2 3 4 5 6 7
ENERGY -3.1244 -.5032 .0000 .0000 .0000 .0000
OCC. NO. 2.0000 2.0000 .0000 .0000 .0000 .0000
1 NI 2px -.0001 .0001 .0015 .0018 .0012 -.0004
2 NI 2px -.9999 .0056 .0213 .0349 .0235 -.0054
3 NI 2px -.0062 -.0140 .1244 -.3887 .2021 -.0182
4 NI 2px .0042 .0037 .0893 .8855 -.0520 .0356
5 NI 3d1+ .0053 .9993 .0268 .0329 .0586 .0005
6 NI 3d1+ -.0002 -.0211 -.5975 .1616 .1313 .0044
7 NI 3d1+ -.0012 -.0159 .7930 .0733 .0616 .0023
8 NI 4f1+ .0013 -.0049 .0117 .1257 1.0211 -.0085
9 NI 4f3+ -.0064 .0000 -.0003 -.0394 .0132 .9991
10 H 2px -.0008 .0024 -.0974 -.1614 -.2576 -.0029
11 H 2px .0003 -.0057 -.2060 -.2268 -.0768 -.0079
Molecular orbitals for symmetry species 3
ORBITAL 2 3 4 5 6 7
ENERGY -3.1244 -.5032 .0000 .0000 .0000 .0000
OCC. NO. 2.0000 2.0000 .0000 .0000 .0000 .0000
1 NI 2py -.0001 .0001 -.0015 .0018 .0012 .0004
2 NI 2py -.9999 .0056 -.0213 .0349 .0235 .0054
3 NI 2py -.0062 -.0140 -.1244 -.3887 .2021 .0182
4 NI 2py .0042 .0037 -.0893 .8855 -.0520 -.0356
5 NI 3d1- .0053 .9993 -.0268 .0329 .0586 -.0005
6 NI 3d1- -.0002 -.0211 .5975 .1616 .1313 -.0044
7 NI 3d1- -.0012 -.0159 -.7930 .0733 .0616 -.0023
8 NI 4f3- .0064 .0000 -.0003 .0394 -.0132 .9991
9 NI 4f1- .0013 -.0049 -.0117 .1257 1.0211 .0085
10 H 2py -.0008 .0024 .0974 -.1614 -.2576 .0029
11 H 2py .0003 -.0057 .2060 -.2268 -.0768 .0079
Molecular orbitals for symmetry species 4
ORBITAL 1 2 3 4
ENERGY -.0799 .0000 .0000 .0000
OCC. NO. 1.0000 .0000 .0000 .0000
1 NI 3d2- -.9877 -.0969 .0050 -.1226
2 NI 3d2- -.1527 .7651 .0019 .6255
3 NI 3d2- -.0332 -.6365 -.0043 .7705
4 NI 4f2- .0051 -.0037 1.0000 .0028
.. NOTE: contains a nbsp
In difficult situations it can be useful to employ the :kword:`AUFBau` option
of the :program:`SCF` program. Including this option, the subsequent
classification of the orbitals in the different symmetry representations
can be avoided. The program will look for the lowest-energy solution and will
provide with a final occupation. This option must be used with caution. It
is only expected to work in clear closed-shell situations.
We have only printed the orbitals most relevant to the following discussion.
Starting with symmetry 1 (|ao|) we observe that the orbitals
are not mixed at all. Using a basis set contracted to :math:`\ce{Ni}` 5s4p3d1f / :math:`\ce{H}` 3s2p
in symmetry |ao| we obtain 18 :math:`\sigma` molecular orbitals (combinations
from eight atomic |s| functions,
six |pz| functions, three |dzt| functions, and one |fztt| function)
and four :math:`\delta` orbitals (from three |dxtyt| functions and one |fz|
function). Orbitals 6, 10, 13, and 18 are formed by contributions from
the three |dxtyt| and one |fz| :math:`\delta` functions, while the
contributions of the remaining harmonics are zero. These orbitals are :math:`\delta` orbitals
and should not mix with the remaining |ao| orbitals.
The same situation occurs in symmetries |bo| and |bt| (2 and 3) but in this case
we observe an important mixing among the orbitals. Orbitals 7\ |bo| and 7\ |bt|
have main contributions from the harmonics 4f3+ (|fxtt|) and 4f3\ |-| (|fytt|),
respectively. They should be pure
:math:`\phi` orbitals and not mix at all with the remaining :math:`\pi` orbitals.
The first step is to evaluate the importance of the mixings
for future calculations. Strictly, any kind of mixing should be avoided.
If g functions are used, for instance, new contaminations show up. But,
undoubtedly, not all mixings are going to be equally important. If the
rotations occur among occupied or active orbitals the influence
on the results is going to be larger than if they are high secondary
orbitals. :math:`\ce{NiH}` is one of these cases. The ground state of the molecule
is :math:`^2\Delta`. It has two components and we can therefore compute it
by placing the single electron in the |dxy| orbital (leading to a
state of |at| symmetry in |Ctv|) or in the |dxtyt| orbital of the
|ao| symmetry. Both are :math:`\delta` orbitals and the resulting states
will have the same energy provided that no mixing happens. In the
|at| symmetry no mixing is possible because it is only composed
of :math:`\delta` orbitals but in |ao| symmetry the :math:`\sigma` and :math:`\delta` orbitals
can rotate. It is clear that this type of mixing will be more
important for the calculation than the mixing of :math:`\pi` and :math:`\phi`
orbitals. However it might be necessary to prevent it. Because in the
SCF calculation no high symmetry restriction was imposed on the orbitals,
orbitals 2 and 4
of the |bo| and |bt| symmetries have erroneous contributions of
the 4f3+ and 4f3\ |-| harmonics, and they are occupied or active
orbitals in the following CASSCF calculation.
.. index::
single: Option; Supersymmetry
single: Symmetry; Supersymmetry
single: Program; RASSCF
single: RASSCF; Supersymmetry
To use the supersymmetry (:kword:`SUPSym`) option we must
start with proper orbitals. In this case the |ao| orbitals are
symmetry adapted (within the printed accuracy) but not the
|bo| and |bt| orbitals. Orbitals 7\ |bo| and 7\ |bt|
must have zero coefficients for all the harmonics except for
4f3+ and 4f3\ |-|, respectively. The remaining orbitals of these
symmetries (even those not shown) must have zero in the
coefficients corresponding to 4f3+ or 4f3\ |-|. To clean the orbitals
the option :kword:`CLEAnup` of the :program:`RASSCF` program can be used.
Once the orbitals are properly symmetrized we can perform CASSCF
calculations on different electronic states. Deriving the types of the
molecular electronic states resulting from the electron configurations
is not simple in many cases. In general, for a given electronic
configuration several electronic states of the molecule will result.
Wigner and Witmer derived rules for determining what types of molecular
states result from given states of the separated atoms.
In chapter VI of reference :cite:`Herzberg:66` it is possible to
find the tables of the resulting electronic states once the
different couplings and the Pauli principle have been applied.
.. index::
single: Active space
single: Ground state
In the present CASSCF calculation we have chosen the active
space (3d, 4d, :math:`\sigma`, :math:`\sigma^*`) with all the 11 valence
electrons active. If we consider 4d and :math:`\sigma^*` as weakly occupied
correlating orbitals, we are left with 3d and :math:`\sigma` (six orbitals),
which are to be occupied with 11 electrons. Since the bonding
orbital :math:`\sigma` (composed mainly of :math:`\ce{Ni}` 4s and :math:`\ce{H}` 1s) will be doubly
occupied in all low lying electronic states, we are left with nine
electrons to occupy the 3d orbitals. There is thus one hole, and
the possible electronic states are: :math:`^2\Sigma^+`, :math:`^2\Pi`, and :math:`^2\Delta`,
depending on the orbital where the hole is located. Taking :numref:`tab:cc`
into account we observe that we have two low-lying electronic states
in symmetry 1 (:math:`A_1`): :math:`^2\Sigma^+` and :math:`^2\Delta`, and one in each of
the other three symmetries: :math:`^2\Pi` in symmetries 2 (:math:`B_1`) and 3 (:math:`B_2`),
and :math:`^2\Delta` in symmetry 4 (:math:`A_2`). It is not immediately obvious which
of these states is the ground state as they are close in energy. It may
therefore be necessary to study all of them. It has been found at different
levels of theory that the :math:`\ce{NiH}` has a :math:`^2\Delta` ground state :cite:`Pou:94`.
.. index::
single: Excited states; NiH
We continue by computing the :math:`^2\Delta` ground state. The previous SCF
orbitals will be the initial orbitals for the CASSCF calculation. First
we need to know in which |Ctv| symmetry or symmetries we can compute
a :math:`\Delta` state. In the symmetry tables it is determined how the species
of the linear molecules are resolved into those of lower symmetry
(depends also on the orientation of the molecule). In :numref:`tab:cc`
is listed the assignment of the different symmetries for the molecule
placed on the :math:`z` axis.
.. index::
single: Degenerate states
The :math:`\Delta` state has two degenerate components in symmetries |ao| and |at|.
Two CASSCF calculations can be performed, one computing
the first root of |at| symmetry and the second for the first root of |ao| symmetry.
The :program:`RASSCF` input for the state of |at| symmetry would be: ::
&RASSCF &END
Title
NiH 2Delta CAS s, s*, 3d, 3d'.
Symmetry
4
Spin
2
Nactel
11 0 0
Inactive
5 2 2 0
Ras2
6 2 2 2
Thrs
1.0E-07,1.0E-05,1.0E-05
Cleanup
1
4 6 10 13 18
18 1 2 3 4 5 6 7 8 9 10 11 12 16 18 19 20 21 22
4 13 14 15 17
1
1 7
10 1 2 3 4 5 6 7 8 10 11
1 9
1
1 7
10 1 2 3 4 5 6 7 9 10 11
1 8
0
Supsym
1
4 6 10 13 18
1
1 7
1
1 7
0
*Average
*1 2 3
Iter
50,25
LumOrb
End of Input
The corresponding input for symmetry |ao| will be identical except
for the :kword:`SYMMetry` keyword ::
Symmetry
1
.. index::
single: Symmetry Species; C∞v in C2v
.. table:: Resolution of the |Cinfv| species in the |Ctv| species.
:name: tab:cc
====================== ======================
State symmetry |Cinfv| State symmetry |Ctv|
====================== ======================
:math:`\Sigma^+` :math:`A_1`
:math:`\Sigma^-` :math:`A_2`
:math:`\Pi` :math:`B_1 + B_2`
:math:`\Delta` :math:`A_1 + A_2`
:math:`\Phi` :math:`B_1 + B_2`
:math:`\Gamma` :math:`A_1 + A_2`
====================== ======================
.. index::
single: Option; Cleanup
single: Symmetry; Cleanup
single: RASSCF; Cleanup
In the :program:`RASSCF` inputs the :kword:`CLEAnup` option will take the initial orbitals
(SCF here)
and will place zeroes in all the coefficients of orbitals 6, 10, 13, and 18 in symmetry 1,
except in coefficients 13, 14, 15, and 17. Likewise all coefficients 13, 14, 15, and 17
of the remaining |ao| orbitals will be set to zero. The same procedure is used
in symmetries |bo| and |bt|. Once cleaned, and because of the :kword:`SUPSymmetry` option,
the :math:`\delta` orbitals 6, 10, 13, and 18 of |ao| symmetry
will only rotate among themselves and they will not mix with the remaining
|ao| :math:`\sigma` orbitals. The same holds true for :math:`\phi` orbitals 7\ |bo| and 7\ |bt|
in their respective symmetries.
Orbitals can change order during the calculation. |molcas| incorporates a
procedure to check the nature of the orbitals in each iteration. Therefore
the right behavior of the :kword:`SUPSym` option is guaranteed during the
calculation. The procedure can have problems if the initial orbitals are
not symmetrized properly. Therefore, the output with the final results
should be checked to compare the final order of the orbitals and the
final labeling of the :kword:`SUPSym` matrix.
.. index::
single: Option; Average
single: RASSCF; Average option
single: Symmetry; Average
single: Convergence problems; In RASSCF
The :kword:`AVERage` option would average the density matrices of symmetries 2 and 3,
corresponding to the :math:`\Pi` and :math:`\Phi` symmetries in |Cinfv|. In this case
it is not necessary to use the option because the two components of the
degenerate sets in symmetries |bo| and |bt| have the same occupation and
therefore they will have the same shape. The use of the option in a situation
like this (:math:`^2\Delta` and :math:`^2\Sigma^+` states) leads to convergence problems.
The symmetry of the orbitals in symmetries 2 and 3 is retained even if the
:kword:`AVERage` option is not used.
The output for the calculation on symmetry 4 (|at|) contains the following lines: ::
Convergence after 29 iterations
30 2 2 1 -1507.59605678 -.23E-11 3 9 1 -.68E-06 -.47E-05
Wave function printout:
occupation of active orbitals, and spin coupling of open shells (u,d: Spin up or down)
printout of CI-coefficients larger than .05 for root 1
energy= -1507.596057
conf/sym 111111 22 33 44 Coeff Weight
15834 222000 20 20 u0 .97979 .95998
15838 222000 ud ud u0 .05142 .00264
15943 2u2d00 ud 20 u0 -.06511 .00424
15945 2u2d00 20 ud u0 .06511 .00424
16212 202200 20 20 u0 -.05279 .00279
16483 u220d0 ud 20 u0 -.05047 .00255
16485 u220d0 20 ud u0 .05047 .00255
Natural orbitals and occupation numbers for root 1
sym 1: 1.984969 1.977613 1.995456 .022289 .014882 .005049
sym 2: 1.983081 .016510
sym 3: 1.983081 .016510
sym 4: .993674 .006884
.. NOTE: contains a nbsp
.. index::
single: RASSCF; CI coefficients
single: RASSCF; Natural occupation
The state is mainly (weight 96%) described by a single configuration
(configuration number 15834) which placed one electron on the first active
orbital of symmetry 4 (|at|) and the remaining electrons are paired.
A close look to this orbital indicates that is
has a coefficient |-|\.9989 in the first 3d2\ |-| (3\ |dxy|) function and small
coefficients in the other functions. This results clearly indicate that
we have computed the :math:`^2\Delta` state as the lowest root of that symmetry.
The remaining configurations have negligible contributions. If the orbitals
are properly symmetrized, all configurations will be compatible with a
:math:`^2\Delta` electronic state.
The calculation of the first root of symmetry 1 (|ao|) results: ::
Convergence after 15 iterations
16 2 3 1 -1507.59605678 -.19E-10 8 15 1 .35E-06 -.74E-05
Wave function printout:
occupation of active orbitals, and spin coupling of open shells (u,d: Spin up or down)
printout of CI-coefficients larger than .05 for root 1
energy= -1507.596057
conf/sym 111111 22 33 44 Coeff Weight
40800 u22000 20 20 20 -.97979 .95998
42400 u02200 20 20 20 .05280 .00279
Natural orbitals and occupation numbers for root 1
sym 1: .993674 1.977613 1.995456 .022289 .006884 .005049
sym 2: 1.983081 .016510
sym 3: 1.983081 .016510
sym 4: 1.984969 .014882
.. NOTE: contains a nbsp
We obtain the same energy as in the previous calculation. Here the dominant
configuration places one electron on the first active orbital of symmetry 1 (|ao|).
It is important to remember that the orbitals are not ordered by energies or
occupations into the active space. This orbital has also the coefficient |-|\.9989
in the first 3d2\ |-| (3\ |dxtyt|) function. We have then computed the other
component of the :math:`^2\Delta` state. As the :math:`\delta` orbitals in different |Ctv|
symmetries are not averaged
by the program it could happen (not in the present case) that the two energies
differ slightly from each other.
The consequences of not using the :kword:`SUPSym` option are not extremely
severe in the present example. If you perform a calculation without the
option, the obtained energy is: ::
Convergence after 29 iterations
30 2 2 1 -1507.59683719 -.20E-11 3 9 1 -.69E-06 -.48E-05
As it is a broken symmetry solution the energy is lower than in the other
case. This is a typical behavior. If we were using an exact wave function
it would have the right symmetry properties, but approximated wave
functions do not necessarily fulfill this condition. So, more flexibility leads to
lower energy solutions which have broken the orbital symmetry.
If in addition to the :math:`^2\Delta` state we want to compute the lowest :math:`^2\Sigma^+`
state we can use the adapted orbitals from any of the :math:`^2\Delta` state
calculations and use the previous :program:`RASSCF` input without the
:kword:`CLEAnup` option. The orbitals have not changed place in this example.
If they do, one has to change the labels in the :kword:`SUPSym` option.
The simplest way to compute the lowest excited :math:`^2\Sigma^+` state
is having the unpaired electron in one of the :math:`\sigma` orbitals because none of
the other configurations, :math:`\delta^3` or :math:`\pi^3`, leads to the :math:`^2\Sigma^+` term.
However, there are more possibilities such as the configuration
:math:`\sigma^1\sigma^1\sigma^1`; three nonequivalent electrons in three
:math:`\sigma` orbitals. In actuality
the lowest :math:`^2\Sigma^+` state must be computed as a doublet state in symmetry
:math:`A_1`. Therefore, we set the symmetry in the RASSCF to 1 and compute the second
root of the symmetry (the first was the :math:`^2\Delta` state):
.. index::
single: RASSCF; CIroot
single: Excited states
::
CIRoot
1 2
2
Of course the :kword:`SUPSym` option must be maintained.
The use of :kword:`CIROot` indicates that we are computing the second root
of that symmetry. The obtained result: ::
Convergence after 33 iterations
9 2 3 2 -1507.58420263 -.44E-10 2 11 2 -.12E-05 .88E-05
Wave function printout:
occupation of active orbitals, and spin coupling of open shells (u,d: Spin up or down)
printout of CI-coefficients larger than .05 for root 1
energy= -1507.584813
conf/sym 111111 22 33 44 Coeff Weight
40800 u22000 20 20 20 -.97917 .95877
printout of CI-coefficients larger than .05 for root 2
energy= -1507.584203
conf/sym 111111 22 33 44 Coeff Weight
40700 2u2000 20 20 20 .98066 .96169
Natural orbitals and occupation numbers for root 2
sym 1: 1.983492 .992557 1.995106 .008720 .016204 .004920
sym 2: 1.983461 .016192
sym 3: 1.983451 .016192
sym 4: 1.983492 .016204
.. NOTE: contains a nbsp
As we have used two as the dimension of the CI matrix employed in the CI Davidson
procedure we obtain the wave function of two roots, although the optimized
root is the second. Root 1 places one electron in the first active orbital
of symmetry one, which is a 3d2+ (3\ |dxtyt|) :math:`\delta` orbital. Root 2 places
the electron in the second active orbital, which is a :math:`\sigma` orbital with a
large coefficient (.9639) in the first 3d0 (3\ |dzt|) function of the nickel
atom. We have therefore computed the lowest :math:`^2\Sigma^+` state. The two :math:`^2\Sigma^+` states
resulting from the configuration with the three unpaired :math:`\sigma` electrons
is higher in energy at the CASSCF level. If the second root of symmetry |ao|
had not been a :math:`^2\Sigma^+` state we would have to study higher roots of the
same symmetry.
.. index::
single: Orbitals; Active
It is important to remember that the active orbitals are not ordered at all
within the active space. Therefore, their order might vary from calculation
to calculation and, in addition, no conclusions about the orbital energy,
occupation or any other information can be obtained from the
order of the active orbitals.
We can compute also the lowest :math:`^2\Pi` excited state.
The simplest possibility is having the configuration :math:`\pi^3`,
which only leads to one :math:`^2\Pi` state. The unpaired electron
will be placed in either one |bo| or one |bt| orbital. That means
that the state has two degenerate components and we can compute it
equally in both symmetries. There are more possibilities, such as the
configuration :math:`\pi^3\sigma^1\sigma^1` or the configuration :math:`\pi^3\sigma^1\delta^1`.
The resulting :math:`^2\Pi` state will always have two degenerate
components in symmetries
|bo| and |bt|, and therefore it is the wave function analysis which
gives us the information of which configuration leads to
the lowest :math:`^2\Pi` state.
.. index::
single: Convergence problems; In RASSCF
For :math:`\ce{NiH}` it turns out to be non trivial to compute the :math:`^2\Pi` state.
Taking as initial orbitals
the previous SCF orbitals and using any type of restriction such as
the :kword:`CLEAnup`, :kword:`SUPSym` or :kword:`AVERage` options lead to
severe convergence problems like these: ::
45 9 17 1 -1507.42427683 -.65E-02 6 18 1 -.23E-01 -.15E+00
46 5 19 1 -1507.41780710 .65E-02 8 15 1 .61E-01 -.15E+00
47 9 17 1 -1507.42427683 -.65E-02 6 18 1 -.23E-01 -.15E+00
48 5 19 1 -1507.41780710 .65E-02 8 15 1 .61E-01 -.15E+00
49 9 17 1 -1507.42427683 -.65E-02 6 18 1 -.23E-01 -.15E+00
50 5 19 1 -1507.41780710 .65E-02 8 15 1 .61E-01 -.15E+00
No convergence after 50 iterations
51 9 19 1 -1507.42427683 -.65E-02 6 18 1 -.23E-01 -.15E+00
.. NOTE: contains nbsp
The calculation, however, converges in an straightforward way if none of those tools are used: ::
Convergence after 33 iterations
34 2 2 1 -1507.58698677 -.23E-12 3 8 2 -.72E-06 -.65E-05
Wave function printout:
occupation of active orbitals, and spin coupling of open shells (u,d: Spin up or down)
printout of CI-coefficients larger than .05 for root 1
energy= -1507.586987
conf/sym 111111 22 33 44 Coeff Weight
15845 222000 u0 20 20 .98026 .96091
15957 2u2d00 u0 ud 20 .05712 .00326
16513 u220d0 u0 20 ud -.05131 .00263
Natural orbitals and occupation numbers for root 1
sym 1: 1.984111 1.980077 1.995482 .019865 .015666 .004660
sym 2: .993507 .007380
sym 3: 1.982975 .016623
sym 4: 1.983761 .015892
.. NOTE: contains a nbsp
The :math:`\pi` (and :math:`\phi`) orbitals, both in symmetries |bo| and |bt|, are, however,
differently occupied and therefore are not equal as they should be: ::
Molecular orbitals for sym species 2 Molecular orbitals for symmetry species 3
ORBITAL 3 4 ORBITAL 3 4
ENERGY .0000 .0000 ENERGY .0000 .0000
OCC. NO. .9935 .0074 OCC. NO. 1.9830 .0166
1 NI 2px .0001 .0002 1 NI 2py .0018 -.0001
2 NI 2px .0073 .0013 2 NI 2py .0178 -.0002
3 NI 2px -.0155 .0229 3 NI 2py -.0197 -.0329
4 NI 2px .0041 .0227 4 NI 2py .0029 -.0254
5 NI 3d1+ .9990 -.0199 5 NI 3d1- .9998 -.0131
6 NI 3d1+ -.0310 -.8964 6 NI 3d1- .0128 .9235
7 NI 3d1+ -.0105 .4304 7 NI 3d1- .0009 -.3739
8 NI 4f1+ -.0050 .0266 8 NI 4f3- .0001 -.0003
9 NI 4f3+ .0001 .0000 9 NI 4f1- -.0050 -.0177
10 H 2px .0029 -.0149 10 H 2py .0009 .0096
11 H 2px -.0056 -.0003 11 H 2py -.0094 -.0052
.. NOTE: contains a nbsp
Therefore what we have is a symmetry broken solution. To obtain a solution which
is not of broken nature the :math:`\pi` and :math:`\phi` orbitals must be equivalent.
The tool to obtain equivalent orbitals is the :kword:`AVERage` option, which averages
the density matrices of symmetries |bo| and |bt|. But starting with any of the preceding
orbitals and using the :kword:`AVERage` option lead again to convergence problems.
It is necessary to use better initial orbitals; orbitals which have
already equal orbitals in symmetries |bo| and |bt|. One possibility is to perform a
SCF calculation on the :math:`\ce{NiH^+}` cation explicitly indicating occupation one in the two
higher occupied :math:`\pi` orbitals (symmetries 2 and 3):
.. index::
single: SCF; OccNumbers
::
&SCF &END
TITLE
NiH cation
OCCUPIED
8 3 3 1
OCCNO
2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0
2.0 2.0 1.0 <-- Note the extra occupation
2.0 2.0 1.0 <-- Note the extra occupation
2.0
IVO
END OF INPUT
It can take some successive steps to obtain a converged calculation using the
:kword:`CLEAnup`, :kword:`SUPSym`, and :kword:`AVERage` options. The calculation
with a single root did not converge clearly. We obtained, however, a converged
result for the lowest :math:`^2\Pi` state of :math:`\ce{NiH}`
by computing two averaged CASSCF roots and setting a weight of
90% for the first root using the keyword:
.. index::
single: RASSCF; CIroot
single: Option; CIroot
single: RASSCF; Average states
single: Average states
::
CIROot
2 2
1 2
9 1
::
Wave function printout:
occupation of active orbitals, and spin coupling of open shells (u,d: Spin up or down)
printout of CI-coefficients larger than .05 for root 1
energy= -1507.566492
conf/sym 111111 22 33 44 Coeff Weight
4913 222u00 20 d0 u0 -.05802 .00337
15845 222000 u0 20 20 .97316 .94703
15953 2u2d00 u0 20 20 .05763 .00332
16459 2u20d0 u0 20 ud -.05283 .00279
Natural orbitals and occupation numbers for root 1
sym 1: 1.972108 1.982895 1.998480 .028246 .016277 .007159
sym 2: .997773 .007847
sym 3: 1.978019 .016453
sym 4: 1.978377 .016366
.. NOTE: contains a nbsp
The energy of the different states (only the first one shown above) is
printed on the top of their configuration list. The converged energy is
simply an average energy.
The occupation numbers obtained in the section of the :program:`RASSCF` output printed
above are the occupation numbers of the natural orbitals of the corresponding
root. They differ from the occupation numbers printed in the
molecular orbital section where we have pseudonatural molecular orbitals and
average occupation numbers. On top of each of the valence :math:`\pi` orbitals
an average occupation close to 1.5 will be printed; this is a consequence
of the the averaging procedure.
.. index::
single: Natural occupation
single: Orbitals; Natural
The results obtained are only at the CASSCF level. Additional effects have to
be considered and included. The most important of them is the dynamical correlation
effect which can be added by computing, for instance, the CASPT2 energies. The reader can find
a detailed explanation of the different approaches in ref. :cite:`Pou:94`, and a
careful discussion of their consequences and solutions in ref. :cite:`Taylor:92b`.
.. index::
single: Relativistic effects
single: Option; Relint
single: SEWARD; Relint
We are going, however, to point out some details. In the first place the basis set
must include up to g functions for the transition metal atom and up to d
functions for the hydrogen. Relativistic effects must be taken into account,
at least in a simple way as a first order correction. The keyword :kword:`RELInt`
must be then included in the :program:`SEWARD` input to compute the mass-velocity and
one-electron Darwin contact term integrals and obtain a first-order correction
to the energy with respect to relativistic effects at the CASSCF level in the :program:`RASSCF` output.
Scalar relativistic effects can be also included according the Douglas--Kroll
or the Barysz--Sadlej--Snijders transformations, as it will be explained in
:numref:`TUT:sec:SOC`.
The CASPT2 input needed to compute the second-order correction to the energy
will include the number of the CASSCF root to compute. For instance,
for the first root of each symmetry:
.. index::
single: CASPT2
::
&CASPT2 &END
Title
NiH
Frozen
5 2 2 0
Maxit
30
Lroot
1
End of input
.. index::
single: Orbitals; Frozen
single: Option; Frozen
single: CASPT2; Frozen
single: Core; Core correlation
The number of frozen orbitals taken by :program:`CASPT2` will be that specified in the :program:`RASSCF` input
except if this is changed in the :program:`CASPT2` input. In the perturbative step
we have frozen all the occupied orbitals except the active ones. This is motivated by
the desire to include exclusively the dynamical correlation related to the valence
electrons. In this way we neglect correlation between core electrons, named core-core
correlation, and between core and valence electrons, named core-valence correlation.
This is not because the calculation is smaller but because of the inclusion of those
type of correlation in a calculation designed to treat valence correlation is an
inadequate approach. Core-core and core-valence correlation requires additional basis
functions of the same spatial extent as the occupied orbitals being correlated, but
with additional radial and angular nodes. Since the spatial extent of the core
molecular orbitals is small, the exponents of these correlating functions must be
much larger than those of the valence optimized basis sets. The consequence is that
we must avoid the inclusion of the core electrons in the treatment in the first step.
Afterwards, the amount of correlation introduced by the core electrons can be estimated
in separated calculations for the different states and those effects added to the
results with the valence electrons.
.. index::
single: Core; core–valence correlation
Core-valence correlation effects of the 3s and 3p nickel shells can be studied by
increasing the basis set flexibility by uncontracting the
basis set in the appropriate region. There are different possibilities. Here we show
the increase of the basis set by four s, four p, and four d functions. f
functions contribute less to the description of the 3s and 3p shells and can be
excluded. The uncontracted exponents should correspond to the region where the 3s
and 3p shells present their density maximum. Therefore, first we compute the absolute
maxima of the radial distribution of the involved orbitals, then we determine the primitive
gaussian functions which have their maxima in the same region as the orbitals and therefore
which exponents should be uncontracted. The final basis set will be the valence basis set
used before plus the new added functions. In the present example the SEWARD
input can be:
.. index::
single: SEWARD; Inline
single: Basis set; Inline
single: Basis set; Extension
.. extractfile:: advanced/SEWARD.NiH.input
&SEWARD &END
Title
NiH G.S.
Symmetry
X Y
*RelInt
Basis set
Ni.ANO-L...5s4p3d1f.
Ni 0.00000 0.00000 0.000000 Bohr
End of basis
Basis set
Ni....4s4p4d. / Inline
0. 2
* Additional s functions
4 4
3.918870 1.839853 0.804663 0.169846
1. 0. 0. 0.
0. 1. 0. 0.
0. 0. 1. 0.
0. 0. 0. 1.
* Additional p functions
4 4
2.533837 1.135309 0.467891 0.187156
1. 0. 0. 0.
0. 1. 0. 0.
0. 0. 1. 0.
0. 0. 0. 1.
* Additional d functions
4 4
2.551303 1.128060 0.475373 0.182128
1. 0. 0. 0.
0. 1. 0. 0.
0. 0. 1. 0.
0. 0. 0. 1.
Nix 0.00000 0.00000 0.000000 Bohr
End of basis
Basis set
H.ANO-L...3s2p.
H 0.000000 0.000000 2.747000 Bohr
End of basis
End of Input
.. index::
single: Option; Charge
single: SEWARD; Charge
.. compound::
We have used a special format to include the additional functions.
We include the additional 4s4p4d functions for the nickel atom.
The additional basis set input must use a dummy label (``Nix`` here), the
same coordinates of the original atom, and
specify a :kword:`CHARge` equal to zero, whether in an Inline basis set
input as here or by specifically using keyword :kword:`CHARge`. It is not
necessary to include the basis set with the Inline format. A library can
be created for this purpose. In this case the label for the additional
functions could be:
.. index::
single: Basis set; Extension
::
Ni.Uncontracted...4s4p4d. / AUXLIB
Charge
0
.. index::
single: Basis set; Auxiliary libraries
and a proper link to AUXLIB should be included in the script (or in the
input if one uses AUTO).
Now the CASPT2 is going to be different to include also
the correlation related to the 3s,3p shell of the nickel atom. Therefore,
we only freeze the 1s,2s,2p shells: ::
&CASPT2 &END
Title
NiH. Core-valence.
Frozen
3 1 1 0
Maxit
30
Lroot
1
End of input
.. index::
single: BSSE Effect
A final effect one should study is the basis set superposition error (BSSE).
In many cases it is a minor effect but it is an everpresent phenomenon
which should be investigated when high accuracy is required, especially in
determining bond energies, and not only in cases with weakly interacting
systems, as is frequently believed. The most common approach to estimate
this effect is the counterpoise correction: the separated fragment energies
are computed in the total basis set of the system. For a discussion of this
issue see Refs. :cite:`Taylor:92b,Gonzalez:94`. In the present example
we would compute the energy of the isolated nickel atom using a SEWARD input
including the full nickel basis set plus the hydrogen basis set in the
hydrogen position but with the charge set to zero. And then the opposite
should be done to compute the energy of isolated hydrogen. The BSSE depends
on the separation of the fragments and must be
estimated at any computed geometry. For instance, the SEWARD input necessary
to compute the isolated hydrogen atom at a given distance from the ghost
nickel basis set including core uncontracted functions is:
.. index::
single: Basis set; Ghost
.. extractfile:: advanced/BSSE.NiH.sample
>>UNIX mkdir AUXLIB
>>COPY $CurrDir/NiH.NewLib AUXLIB/UNCONTRACTED
&SEWARD &END
Title
NiH. 3s3p + H (BSSE)
Symmetry
X Y
RelInt
Basis set
Ni.ANO-L...5s4p3d1f.
Ni 0.00000 0.00000 0.000000 Bohr
Charge
0.0
End of basis
Basis set
Ni.Uncontracted...4s4p4d. / AUXLIB
Nix 0.00000 0.00000 0.000000 Bohr
Charge
0.0
End of basis
Basis set
H.ANO-L...3s2p.
H 0.000000 0.000000 2.747000 Bohr
End of basis
End of Input
Once the energy of each of the fragments with the corresponding ghost
basis set of the other fragment is determined, the energies of the
completely isolated fragments can be computed and subtracted from those
which have the ghost basis sets. Other approaches used to estimate
the BSSE effect are discussed in Ref. :cite:`Taylor:92b`.
The results obtained at the CASPT2 level are close to those obtained by
MRCI+Q and ACPF treatments but more accurate. They match well with experiment.
The difference is that all the configuration functions (CSFs) of the active
space can be included in CASPT2 in the zeroth-order references for the second-order
perturbation calculation :cite:`Pou:94`, while the other methods have to restrict
the number of configurations.
Calculations of linear molecules become more and more complicated when the
number of unpaired electrons increases. In the following sections we will discuss
the more complicated situation occurring in the :math:`\ce{Ni2}` molecule.
.. _TUT\:sec\:c2:
A diatomic homonuclear molecule: :math:`\ce{C2}`
------------------------------------------------
.. index::
single: Linear molecules; C2
single: C2
:math:`\ce{C2}` is a classical example of a system where near-degeneracy effects have large
amplitudes even near the equilibrium internuclear separation. The biradical
character of the ground state of the molecule suggest that a single
configurational treatment will not be appropriate for accurate descriptions
of the spectroscopic constants :cite:`Roos:87`.
There are two nearly degenerate states: :math:`^1\Sigma_g^+` and :math:`^3\Pi_u`. The latter
was earlier believed to be the ground state, an historical assignment which can
be observed in the traditional labeling of the states.
As :math:`\ce{C2}` is a |Dinfh| molecule, we have to compute it in |Dth| symmetry. We
make a similar analysis as for the |Ctv| case. We begin by
classifying the functions in |Dinfh| in :numref:`tab:dinfh`.
The molecule is placed on the :math:`z` axis.
.. index::
single: Spherical Harmonics; D∞h
.. table:: Classification of the spherical harmonics in the |Dinfh| group\ [#b]_.
:name: tab:dinfh
================ ======= ======= ======= =======
Symmetry Spherical harmonics
================ ===============================
:math:`\sigma_g` |s| |dzt|
:math:`\sigma_u` |pz| |dzt|
:math:`\pi_g` |dxz| |dyz|
:math:`\pi_u` |px| |py| |fx| |fy|
:math:`\delta_g` |dxtyt| |dxy|
:math:`\delta_u` |fxyz| |fz|
:math:`\phi_u` |fxtt| |fytt|
================ ======= ======= ======= =======
.. [#b] Functions placed on the symmetry center.
:numref:`tab:d2h` classifies
the functions and orbitals into the symmetry representations of the |Dth|
symmetry. Note that in :numref:`tab:d2h` subindex :math:`b` stands for bonding combination and
:math:`a` for antibonding combination.
.. index::
single: Spherical Harmonics; D2h
.. table:: Classification of the spherical harmonics and |Dinfh| orbitals in the |Dth| group\ [#c]_.
:name: tab:d2h
============ ====================================== ====================================== ====================================== ====================================== ====================================== ======================================
Symm.\ [#d]_ Spherical harmonics (orbitals in |Dinfh|)
============ =========================================================================================================================================================================================================================================
|ag|\(1) |s|\ :math:`_b` (:math:`\sigma_g`) |pz|\ :math:`_b` (:math:`\sigma_g`) |dzt|\ :math:`_b` (:math:`\sigma_g`) |dxtyt|\ :math:`_b` (:math:`\delta_g`) |fztt|\ :math:`_b` (:math:`\sigma_g`) |fz|\ :math:`_b` (:math:`\delta_g`)
|bttu|\(2) |px|\ :math:`_b` (:math:`\pi_u`) |dxz|\ :math:`_b` (:math:`\pi_u`) |fx|\ :math:`_b` (:math:`\pi_u`) |fxtt|\ :math:`_b` (:math:`\phi_u`)
|btu|\(3) |py|\ :math:`_b` (:math:`\pi_u`) |dyz|\ :math:`_b` (:math:`\pi_u`) |fy|\ :math:`_b` (:math:`\pi_u`) |fytt|\ :math:`_b` (:math:`\phi_u`)
|bog|\(4) |dxy|\ :math:`_b` (:math:`\delta_g`) |fxyz|\ :math:`_b` (:math:`\delta_g`)
|bou|\(5) |s|\ :math:`_a` (:math:`\sigma_u`) |pz|\ :math:`_a` (:math:`\sigma_u`) |dzt|\ :math:`_a` (:math:`\sigma_u`) |dxtyt|\ :math:`_a` (:math:`\delta_u`) |fztt|\ :math:`_a` (:math:`\sigma_u`) |fz|\ :math:`_a` (:math:`\delta_u`)
|btg|\(6) |py|\ :math:`_a` (:math:`\pi_g`) |dyz|\ :math:`_a` (:math:`\pi_g`) |fy|\ :math:`_a` (:math:`\pi_g`) |fytt|\ :math:`_a` (:math:`\phi_g`)
|bttg|\(7) |px|\ :math:`_a` (:math:`\pi_g`) |dxz|\ :math:`_a` (:math:`\pi_g`) |fx|\ :math:`_a` (:math:`\pi_g`) |fxtt|\ :math:`_a` (:math:`\phi_g`)
|au|\(8) |dxy|\ :math:`_a` (:math:`\delta_u`) |fxyz|\ :math:`_a` (:math:`\delta_u`)
============ ====================================== ====================================== ====================================== ====================================== ====================================== ======================================
.. [#c] Subscripts :math:`b` and :math:`a` refer to the bonding and antibonding combination of the AO's, respectively.
.. [#d] In parenthesis the number of the symmetry in |molcas|. Note that the number and order of the
symmetries depend on the generators and the orientation of the molecule.
The order of the symmetries, and therefore the number they have in |molcas|, depends
on the generators used in the :program:`SEWARD` input. This must be carefully checked
at the beginning of any calculation. In addition, the orientation of the molecule on the
cartesian axis can change the labels of the symmetries. In :numref:`tab:d2h` for
instance we have used the order and numbering of a calculation performed with the
three symmetry planes of the |Dth| point group (X Y Z in the :program:`SEWARD` input)
and the :math:`z` axis as the intermolecular axis (that is, :math:`x` and :math:`y` are equivalent in |Dth|).
Any change in the orientation of the molecule will affect the labels of the orbitals
and states. In this case the :math:`\pi` orbitals will belong to the |bttu|, |btu|,
|btg|, and |bttg| symmetries. For instance, with :math:`x` as the intermolecular axis |bttu| and |bttg| will
be replaced by |bou| and |bog|, respectively, and finally with :math:`y` as the intermolecular axis
|bou|, |bttu|, |bttg|, and |bog| would be the :math:`\pi` orbitals.
It is important to remember that |molcas| works with symmetry adapted basis functions.
Only the symmetry independent atoms are required in the :program:`SEWARD` input. The remaining
ones will be generated by the symmetry operators. This is also the case for the
molecular orbitals. |molcas| will only print the coefficients of the symmetry adapted
basis functions.
.. index::
single: Symmetry; Adapted basis functions
The necessary information to obtain the complete set of orbitals
is contained in the SEWARD output. Consider the case of the |ag| symmetry: ::
**************************************************
******** Symmetry adapted Basis Functions ********
**************************************************
Irreducible representation : ag
Basis function(s) of irrep:
Basis Label Type Center Phase Center Phase
1 C 1s0 1 1 2 1
2 C 1s0 1 1 2 1
3 C 1s0 1 1 2 1
4 C 1s0 1 1 2 1
5 C 2pz 1 1 2 -1
6 C 2pz 1 1 2 -1
7 C 2pz 1 1 2 -1
8 C 3d0 1 1 2 1
9 C 3d0 1 1 2 1
10 C 3d2+ 1 1 2 1
11 C 3d2+ 1 1 2 1
12 C 4f0 1 1 2 -1
13 C 4f2+ 1 1 2 -1
.. NOTE: contains a nbsp
The previous output indicates that symmetry adapted basis function 1,
belonging to the |ag| representation, is formed by
the symmetric combination of a s type function centered on atom C and
another s type function centered on the redundant center 2, the second
carbon atom. Combination s+s constitutes a bonding :math:`\sigma_g`\-type
orbital. For the |pz| function however the combination must be
antisymmetric. It is the only way to make the |pz| orbitals overlap
and form a bonding orbital of |ag| symmetry. Similar combinations are obtained for the
remaining basis sets of the |ag| and other symmetries.
The molecular orbitals will be combinations of these symmetry adapted
functions. Consider the |ag| orbitals: ::
SCF orbitals
Molecular orbitals for symmetry species 1
ORBITAL 1 2 3 4 5 6
ENERGY -11.3932 -1.0151 -.1138 .1546 .2278 .2869
OCC. NO. 2.0000 2.0000 .0098 .0000 .0000 .0000
1 C 1s0 1.4139 -.0666 -.0696 .2599 .0626 .0000
2 C 1s0 .0003 1.1076 -.6517 1.0224 .4459 .0000
3 C 1s0 .0002 -.0880 -.2817 .9514 .0664 .0000
4 C 1s0 .0000 -.0135 -.0655 .3448 -.0388 .0000
5 C 2pz -.0006 -.2581 -1.2543 1.1836 .8186 .0000
6 C 2pz .0000 .1345 -.0257 2.5126 1.8556 .0000
7 C 2pz .0005 -.0192 -.0240 .7025 .6639 .0000
8 C 3d0 .0003 .0220 -.0005 -.9719 .2430 .0000
9 C 3d0 -.0001 -.0382 -.0323 -.8577 .2345 .0000
10 C 3d2+ .0000 .0000 .0000 .0000 .0000 -.7849
11 C 3d2+ .0000 .0000 .0000 .0000 .0000 -.7428
12 C 4f0 -.0002 -.0103 -.0165 .0743 .0081 .0000
13 C 4f2+ .0000 .0000 .0000 .0000 .0000 -.0181
.. NOTE: contains a nbsp
In |molcas| outputs only 13 coefficients for orbital are going to be printed
because they are the coefficients of the symmetry adapted basis
functions. If the orbitals were not composed by symmetry adapted basis
functions they would have, in this case, 26 coefficients, two for type of
function (following the scheme observed above in the :program:`SEWARD` output),
symmetrically combined the s and d functions and antisymmetrically
combined the p and f functions.
To compute |Dinfh| electronic states using the |Dth| symmetry we need
to go to the symmetry tables and determine how the species
of the linear molecules are resolved into those of lower symmetry
(this depends also on the orientation of the molecule :cite:`Herzberg:66`).
:numref:`tab:dd` lists the case of a |Dinfh| linear molecule with :math:`z` as
the intermolecular axis.
.. index::
single: Symmetry; D∞h in D2h
.. table:: Resolution of the |Dinfh| species in the |Dth| species.
:name: tab:dd
======================== ========================
State symmetry |Dinfh| State symmetry |Dth|
======================== ========================
:math:`\Sigma^+_g` :math:`A_g`
:math:`\Sigma^+_u` :math:`B_{1u}`
:math:`\Sigma^-_g` :math:`B_{1g}`
:math:`\Sigma^-_u` :math:`A_u`
:math:`\Pi_g` :math:`B_{2g} + B_{3g}`
:math:`\Pi_u` :math:`B_{2u} + B_{3u}`
:math:`\Delta_g` :math:`A_{g} + B_{1g}`
:math:`\Delta_u` :math:`A_{u} + B_{1u}`
:math:`\Phi_g` :math:`B_{2g} + B_{3g}`
:math:`\Phi_u` :math:`B_{2u} + B_{3u}`
:math:`\Gamma_g` :math:`A_{g} + B_{1g}`
:math:`\Gamma_u` :math:`A_{u} + B_{1u}`
======================== ========================
.. index::
single: Ground state
single: RASSCF; Supersymmetry
single: Option; Supersymmetry
single: Symmetry; Supersymmetry
To compute the ground state of :math:`\ce{C2}`, a :math:`^1\Sigma_g^+` state, we will
compute a singlet state of symmetry :math:`A_g` (1 in this context).
The input files for a CASSCF calculation on the :math:`\ce{C2}` ground state
will be:
.. extractfile:: advanced/RASSCF.supersymmetry.input
&SEWARD &END
Title
C2
Symmetry
X Y Z
Basis set
C.ANO-L...4s3p2d1f.
C .00000000 .00000000 1.4
End of basis
End of input
&SCF &END
Title
C2
ITERATIONS
40
Occupied
2 1 1 0 2 0 0 0
End of input
&RASSCF &END
Title
C2
Nactel
4 0 0
Spin
1
Symmetry
1
Inactive
2 0 0 0 2 0 0 0
Ras2
1 1 1 0 1 1 1 0
*Average
*2 2 3 6 7
Supsymmetry
1
3 6 9 11
1
1 6
1
1 6
0
1
3 5 8 12
1
1 6
1
1 6
0
Iter
50,25
Lumorb
End of input
In this case the SCF orbitals are already clean symmetry adapted orbitals
(within the printed accuracy).
We can then directly use the :kword:`SUPSym` option. In symmetries
|ag| and |bou| we restrict the rotations among the :math:`\sigma` and
the :math:`\delta` orbitals, and in symmetries |bttu|, |btu|, |btg|, and
|bttg| the rotations among :math:`\pi` and :math:`\phi` orbitals. Additionally,
symmetries |bttu| and |btu| and symmetries |btg| and
|bttg| are averaged, respectively, by using
the :kword:`AVERage` option. They belong to the :math:`\Pi_u` and
:math:`\Pi_g` representations in |Dinfh|, respectively.
A detailed explanation on different CASSCF calculations on the :math:`\ce{C2}`
molecule and their states can be found elsewhere :cite:`Roos:87`.
Instead we include here an example of how to combine the use of
UNIX shell script commands with |molcas| as a powerful tool.
The following example computes the transition dipole moment for the transition
from the :math:`^1\Sigma_g^+` state to the :math:`^1\Pi_u` state in the :math:`\ce{C2}`
molecule. This transition is known as the Phillips bands :cite:`Herzberg:66`.
This is not a serious attempt to compute this property accurately, but serves
as an example of how to set up an automatic calculation.
The potential curves are computed using CASSCF wavefunctions
along with the transition dipole moment.
.. index::
single: Excited states; C2
Starting orbitals are generated by computing a CI wavefunction once and
using the natural orbitals. We loop over a set of distances, compute the
CASSCF wave functions for both states and use :program:`RASSI` to compute the TDMs.
Several UNIX commands are used to manipulate input and output files,
such as grep, sed, and the *awk* language. For instance, an explicit "sed"
is used to insert the geometry into the seward input; the final CASSCF
energy is extracted with an explicit "grep", and the TDM is extracted from the
RASSI output using an *awk* script. We are not going to include the *awk* scripts
here. Other tools can be used to obtain and collect the data.
In the first script, when the loop over geometries is done, four files are available:
geom.list (contains the distances), tdm.list (contains the TDMs),
e1.list (contains the energy for the :math:`^1\Sigma_g^+` state), and
e2.list (contains the energy for the :math:`^1\Pi_u` state). In the second script the vibrational
wave functions for the two states and the vibrationally averaged TDMs
are now computed using the :program:`VIBROT` program. We will retain the RASSCF outputs
in the scratch directory to check the wave function. It is always dangerous
to assume that the wave functions will be correct in a CASSCF calculation.
Different problems such as root flippings or incorrect orbitals rotating into the
active space are not uncommon. Also, it is always necessary to control that the
CASSCF calculation has converged. The first script (Korn shell) is:
.. index::
single: Shell script
::
#!/bin/ksh
#
# perform some initializations
#
export Project='C2'
export WorkDir=/temp/$LOGNAME/$Project
export Home=/u/$LOGNAME/$Project
echo "No log" > current.log
trap 'cat current.log ; exit 1' ERR
mkdir $WorkDir
cd $WorkDir
#
# Loop over the geometries and generate input for vibrot
#
list="1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 5.0 10.0"
scf='yes'
print "Sigma" > e1.list
print "Pi" > e2.list
for geom in $list
do
#--- run seward
print "Dist $geom" >> geom.list
sed -e "s/#/$geom/" $Home/$Project.seward.input > seward.input
molcas seward.input > current.log
#--- optionally run scf, motra, guga and mrci to obtain good starting orbitals
if [ "$scf" = 'yes' ]
then
scf='no'
molcas $Home/$Project.scf.input > current.log
molcas $Home/$Project.motra.input > current.log
molcas $Home/$Project.guga.input > current.log
molcas $Home/$Project.mrci.input > current.log
cp $Project.CiOrb $Project.RasOrb1
cp $Project.CiOrb $Project.RasOrb2
fi
#--- rasscf wavefunction for 1Sg+
ln -fs $Project.Job001 JOBIPH
ln -fs $Project.RasOrb1 INPORB
molcas $Home/$Project.rasscf1.input > current.log
cat current.log >> rasscf1.log
cat current.log | grep -i 'average ci' >> e1.list
cp $Project.RasOrb $Project.RasOrb1
rm -f JOBIPH INPORB
#--- rasscf wavefunction for 1Pu
ln -fs $Project.Job002 JOBIPH
ln -fs $Project.RasOrb2 INPORB
molcas $Home/$Project.rasscf2.input > current.log
cat current.log >> rasscf2.log
cat current.log | grep -i 'average ci' >> e2.list
cp $Project.RasOrb $Project.RasOrb2
rm -f JOBIPH INPORB
#--- rassi to obtain transition
ln -fs $Project.Job001 JOB001
ln -fs $Project.Job002 JOB002
molcas $Home/$Project.rassi.input > current.log
awk -f $Home/tdm.awk current.log >> tdm.list
rm -f JOB001 JOB002
#---
done
#
# Finished so clean up the files.
#
print "Calculation finished" >&2
cd -
rm $WorkDir/molcas.temp*
#rm -r $WorkDir
exit 0
In a second script we will compute the vibrational wave functions ::
#!/bin/ksh
#
# perform some initializations
#
export Project='C2'
export WorkDir=/temp/$LOGNAME/$Project
export Home=/u/$LOGNAME/$Project
echo "No log" > current.log
trap 'cat current.log ; exit 1' ERR
mkdir $WorkDir
cd $WorkDir
#
# Build vibrot input
#
cp e1.list $Home
cp e2.list $Home
cp geom.list $Home
cp tdm.list $Home
#---
cat e1.list geom.list | awk -f $Home/wfn.awk > vibrot1.input
cat e2.list geom.list | awk -f $Home/wfn.awk > vibrot2.input
cat tdm.list geom.list | awk -f $Home/tmc.awk > vibrot3.input
#---
ln -fs $Project.VibWvs1 VIBWVS
molcas vibrot1.input > current.log
cat current.log
rm -f VIBWVS
#---
ln -fs $Project.VibWvs2 VIBWVS
molcas vibrot2.input > current.log
cat current.log
rm -f VIBWVS
#---
ln -fs $Project.VibWvs1 VIBWVS1
ln -fs $Project.VibWvs2 VIBWVS2
molcas vibrot3.input > current.log
cat current.log
rm -f VIBWVS1 VIBWVS2
#
# Finished so clean up the files.
#
print "Calculation finished" >&2
cd -
rm $WorkDir/molcas.temp*
#rm -r $WorkDir
exit 0
The input for the first part of the calculations include the
SEWARD, SCF, MOTRA, GUGA, and MRCI inputs:
.. index::
single: SEWARD
single: SCF
single: MOTRA
single: GUGA
single: MRCI
.. extractfile:: advanced/MRCI.C2.input
&SEWARD &END
Title
C2
Pkthre
1.0D-11
Symmetry
X Y Z
Basis set
C.ANO-S...3s2p.
C .00000000 .00000000 #
End of basis
End of input
&SCF &END
Title
C2
ITERATIONS
40
Occupied
2 1 1 0 2 0 0 0
End of input
&MOTRA &END
Title
C2 molecule
Frozen
1 0 0 0 1 0 0 0
LumOrb
End of input
&GUGA &END
Title
C2 molecule
Electrons
8
Spin
1
Inactive
1 1 1 0 1 0 0 0
Active
0 0 0 0 0 0 0 0
CiAll
1
End of Input
&MRCI &END
Title
C2 molecule
SDCI
End of input
We are going to use a small ANO [3s2p] basis set because our purpose
it is not to obtain an extreme accuracy.
In the SEWARD input the sign "#" will be replaced by the right distance
using the "sed" command. In the MOTRA input we have frozen the two
core orbitals in the molecule, which will be recognized by the MRCI
program. The GUGA input defines the reference space of configurations
for the subsequent MRCI or ACPF calculation. In this case the
valence orbitals are doubly occupied and there is only one reference configuration
(they are included as inactive). We thus use one single
configuration to perform the SDCI calculation and obtain the initial
set of orbitals for the CASSCF calculation.
The lowest :math:`^1\Sigma_g^+` state in :math:`\ce{C2}` is the result of the
electronic configuration [core]\ :math:`(2\sigma_g)^2(2\sigma_u)^2(1\pi_u)^4`.
Only one electronic state is obtained from this
configuration. The configuration :math:`(1\pi_u)^3(3\sigma_g)^1`
is close in energy and generates two possibilities,
one :math:`^3\Pi_u` and one :math:`^1\Pi_u` state. The former is the lowest
state of the Swan bands, and was thought to be the ground state of
the molecule. Transitions to the :math:`^1\Pi_u` state are known as the
Phillips band and this is the state we are going to compute.
We have the possibility to compute the state in symmetry |bttu| or
|btu| (|molcas| symmetry groups 2 and 3, respectively) in the |Dth|
group, because both represent the degenerate :math:`\Pi_u` symmetry in |Dinfh|.
.. index::
single: RASSCF
single: RASSCF; Average option
single: Option; Average
single: Excited states
The RASSCF input file to compute the two states are: ::
&RASSCF &END
Title
C2 1Sigmag+ state.
Nactel
4 0 0
Spin
1
Symmetry
1
Inactive
2 0 0 0 2 0 0 0
Ras2
1 1 1 0 1 1 1 0
*Average
*2 2 3 6 7
OutOrbitals
Natural
1
Iter
50,25
Lumorb
End of input
&RASSCF &END
Title
C2 1Piu state.
Nactel
4 0 0
Spin
1
Symmetry
2
Inactive
2 0 0 0 2 0 0 0
Ras2
1 1 1 0 1 1 1 0
Average
2 2 3 6 7
OutOrbitals
Natural
1
Iter
50,25
Lumorb
End of input
We can skip the :kword:`SUPSym` option because our basis set
contains only s,p functions and no undesired rotations can
happen. Symmetries |bttu| and |btu| on one hand and |btg| and |bttg| on
the other are averaged. Notice that to obtain natural orbitals we have
used keyword :kword:`OUTOrbitals` instead of the old :program:`RASREAD`
program. In addition, we need the :program:`RASSI` input:
.. index::
single: RASSCF
single: OUTORBITALS; Natural orbitals
single: Orbitals; Natural
single: RASSI
single: Properties; Transition dipole moments
single: Transition dipole moments
::
&RASSI &END
NrOfJobiphs
2 1 1
1
1
End of input
.. index::
single: Program; VibRot
single: VibRot; Potential
single: VibRot; Orbital
single: VibRot; Vibwvs
The :program:`VIBROT` inputs to compute the vibrational--rotational
analysis and spectroscopic constants of the state should be: ::
&VIBROT &END
RoVibrational spectrum
Title
Vib-Rot spectrum for C2. 1Sigmag+
Atoms
0 C 0 C
Grid
400
Range
2.0 10.0
Vibrations
3
Rotations
0 4
Orbital
0
Potential
2.2 -75.42310136
...
End of input
Under the keyword :kword:`POTEntial` the bond distance and
potential energy (both in au) of the corresponding state
must be included. In this case we are going to compute three
vibrational quanta and four rotational quantum numbers.
For the :math:`^1\Pi_u` state, the keyword :kword:`ORBItal` must be
set to one, corresponding to the orbital angular momentum
of the computed state. :program:`VIBROT` fits the potential curve to
an analytical curve using splines. The ro-vibrational
Schrödinger equation is then solved numerically (using
Numerov's method) for one vibrational state at a time and
for the specified number of rotational quantum numbers.
File :file:`VIBWVS` will contain the corresponding wave
function for further use.
.. index::
single: Diatomic molecules
single: Spectroscopy
single: Properties; Spectroscopic
.. compound::
Just to give some of the results obtained, the spectroscopic
constants for the :math:`^1\Sigma_g^+` state were: ::
Re(a) 1.4461
De(ev) 3.1088
D0(ev) 3.0305
we(cm-1) .126981E+04
wexe(cm-1) -.130944E+02
weye(cm-1) -.105159E+01
Be(cm-1) .134383E+01
Alphae(cm-1) .172923E-01
Gammae(cm-1) .102756E-02
Dele(cm-1) .583528E-05
Betae(cm-1) .474317E-06
and for the :math:`^1\Pi_u` state: ::
Re(a) 1.3683
De(ev) 2.6829
D0(ev) 2.5980
we(cm-1) .137586E+04
wexe(cm-1) -.144287E+02
weye(cm-1) .292996E+01
Be(cm-1) .149777E+01
Alphae(cm-1) .328764E-01
Gammae(cm-1) .186996E-02
Dele(cm-1) .687090E-05
Betae(cm-1) -.259311E-06
.. index::
single: Properties; Vibrationally averaged TDMs
single: Properties; Lifetimes
single: Diatomic molecules; Vibrationally averaged TDMs
single: Diatomic molecules; Lifetimes
single: VibRot; Observable
To compute vibrationally averaged TDMs the :program:`VIBROT` input must be: ::
&VIBROT &END
Transition moments
Observable
Transition dipole moment
2.2 0.412805
...
End of input
.. compound::
Keyword :kword:`OBSErvable` indicates the start of input
for radial functions of observables other than the energy.
In the present case the vibrational--rotational matrix elements
of the transition dipole moment function will be generated.
The values of the bond distance and the TDM at each distance
must be then included in the input. VIBROT also requires
the :file:`VIBWVS1` and :file:`VIBWVS2` files
containing the vibrational wave functions of the involved electronic states.
The results obtained contain matrix elements, transition
moments over vibrational wave functions, and the lifetimes of the
transition among all the computed vibrational--rotational states.
The radiative lifetime of a vibrational level depends on the
sum of the transition probabilities to all lower vibrational
levels in all lower electronic states. If rotational effects are
neglected, the lifetime (:math:`\tau_v'`) can be written as
.. math:: \tau_v' = \left( \sum_{v''} A_{v'v''} \right)^{-1}
where :math:`v'` and :math:`v''` are the vibrational
levels of the lower and upper electronic state and :math:`A_{v'v''}` is the
Einstein :math:`A` coefficient (ns\ :math:`^{-1}`) computed as
.. math:: A_{v'v''} = 21.419474\, (\Delta E_{v'v''})^3 (\text{TDM}_{v'v''})^2
:math:`\Delta E_{v'v''}` is the energy difference (au) and :math:`\text{TDM}_{v'v''}`
the transition dipole moment (au) of the transition.
For instance, for rotational states zero of the :math:`^1\Sigma^+_g` state
and one of the :math:`^1\Pi_u` state: ::
Rotational quantum number for state 1: 0, for state 2: 1
--------------------------------------------------------------------------------
Overlap matrix for vibrational wave functions for state number 1
1 1 .307535 2 1 .000000 2 2 .425936 3 1 .000000 3 2 .000000 3 3 .485199
Overlap matrix for vibrational wave functions for state number 2
1 1 .279631 2 1 .000000 2 2 .377566 3 1 .000000 3 2 .000000 3 3 .429572
Overlap matrix for state 1 and state 2 functions
-.731192 -.617781 -.280533
.547717 -.304345 -.650599
-.342048 .502089 -.048727
Transition moments over vibrational wave functions (atomic units)
-.286286 -.236123 -.085294
.218633 -.096088 -.240856
-.125949 .183429 .005284
Energy differences for vibrational wave functions(atomic units)
1 1 .015897 2 1 .010246 2 2 .016427 3 1 .004758 3 2 .010939 3 3 .017108
Contributions to inverse lifetimes (ns-1)
No degeneracy factor is included in these values.
1 1 .000007 2 1 .000001 2 2 .000001 3 1 .000000 3 2 .000001 3 3 .000000
Lifetimes (in nano seconds)
v tau
1 122090.44
2 68160.26
3 56017.08
.. NOTE: contains a nbsp
Probably the most important caution when using the VIBROT program
in diatomic molecules is that the number of vibrational
states to compute and the accuracy obtained depends
strongly on the computed surface. In the present case we
compute all the curves to the dissociation limit. In other cases, the program
will complain if we try to compute states which lie at energies
above those obtained in the calculation of the curve.
.. index::
single: Ni2
single: Linear molecules; Ni2
.. _TUT\:sec\:ni2:
A transition metal dimer: :math:`\ce{Ni2}`
------------------------------------------
This section is a brief comment on a complex situation in a diatomic
molecule such as :math:`\ce{Ni2}`. Our purpose is to compute the ground state
of this molecule. An explanation of how to calculate it accurately can
be found in ref. :cite:`Pou:94`. However we will concentrate on computing
the electronic states at the CASSCF level.
The nickel atom has two close low-lying configurations 3\ :math:`d^8`\4s\ :math:`^2` and
3\ :math:`d^9`\4\ :math:`s^1`. The combination of two neutral :math:`\ce{Ni}` atoms leads to a
:math:`\ce{Ni2}` dimer whose ground state has been somewhat controversial.
For our purposes we commence with the assumption that it is
one of the states derived
from 3d\ :math:`^9`\4\ :math:`s^1` :math:`\ce{Ni}` atoms, with a single bond between the 4s orbitals,
little 3d involvement, and the holes localized in the 3d\ :math:`\delta` orbitals.
Therefore, we compute the states resulting from two
holes on :math:`\delta` orbitals: :math:`\delta\delta` states.
.. index::
single: Transition metals
single: Excited states; Ni2
We shall not go through the procedure leading to the different electronic
states that can arise from these electronic configurations, but refer to
the Herzberg book on diatomic molecules :cite:`Herzberg:66` for details. In
|Dinfh| we have three possible configurations with two holes, since the
:math:`\delta` orbitals can be either *gerade* (:math:`g`) or *ungerade* (:math:`u`):
:math:`(\delta_g)^{-2}`, :math:`(\delta_g)^{-1}(\delta_u)^{-1}`, or :math:`(\delta_u)^{-2}`.
The latter situation corresponds to nonequivalent electrons while the other
two to equivalent electrons.
Carrying through the analysis we obtain the following electronic states:
| :math:`(\delta_g)^{-2}`: :math:`^1\Gamma_g`, :math:`^3\Sigma_g^-`, :math:`^1\Sigma_g^+`
| :math:`(\delta_u)^{-2}`: :math:`^1\Gamma_g`, :math:`^3\Sigma_g^-`, :math:`^1\Sigma_g^+`
| :math:`(\delta_g)^{-1}(\delta_u)^{-1}`: :math:`^3\Gamma_u`, :math:`^1\Gamma_u`, :math:`^3\Sigma_u^+`,
:math:`^3\Sigma_u^-`, :math:`^1\Sigma_u^+`, :math:`^1\Sigma_u^-`
In all there are thus 12 different electronic states.
.. compound::
Next, we need to classify these electronic states in the lower symmetry
|Dth|, in which |molcas| works. This is done in :numref:`tab:dd`, which
relates the symmetry in |Dinfh| to that of |Dth|. Since we have only
:math:`\Sigma^+`, :math:`\Sigma^-`, and :math:`\Gamma` states here, the |Dth| symmetries
will be only :math:`A_g`, :math:`A_u`, :math:`B_{1g}`, and :math:`B_{1u}`. The table above can
now be rewritten in |Dth|:
| :math:`(\delta_g)^{-2}`: (|SAG| + |SBOG|), |TBOG|, |SAG|
| :math:`(\delta_u)^{-2}`: (|SAG| + |SBOG|), |TBOG|, |SAG|
| :math:`(\delta_g)^{-1}(\delta_u)^{-1}`: (|TAU| + |TBOU|), (|SAU| + |SBOU|),
|TBOU|, |TAU|, |SBOU|, |SAU|
or, if we rearrange the table after the |Dth| symmetries:
| |SAG|: :math:`^1\Gamma_g(\delta_g)^{-2}`, :math:`^1\Gamma_g(\delta_u)^{-2}`,
:math:`^1\Sigma_g^+(\delta_g)^{-2}`, :math:`^1\Sigma_g^+(\delta_u)^{-2}`
| |SBOU|: :math:`^1\Gamma_u(\delta_g)^{-1}(\delta_u)^{-1}`,
:math:`^1\Sigma_u^+(\delta_g)^{-1}(\delta_u)^{-1}`
| |SBOG|: :math:`^1\Gamma_g(\delta_g)^{-2}`, :math:`^1\Gamma_g(\delta_u)^{-2}`
| |SAU|: :math:`^1\Gamma_u(\delta_g)^{-1}(\delta_u)^{-1}`,
:math:`^1\Sigma_u^-(\delta_g)^{-1}(\delta_u)^{-1}`
| |TBOU|: :math:`^3\Gamma_u(\delta_g)^{-1}(\delta_u)^{-1}`,
:math:`^3\Sigma_u^+(\delta_g)^{-1}(\delta_u)^{-1}`
| |TBOG|: :math:`^3\Sigma_g^-(\delta_g)^{-2}`, :math:`^3\Sigma_g^-(\delta_u)^{-2}`
| |TAU|: :math:`^3\Gamma_u(\delta_g)^{-1}(\delta_u)^{-1}`,
:math:`^3\Sigma_u^-(\delta_g)^{-1}(\delta_u)^{-1}`
It is not necessary to compute all the states because some of
them (the :math:`\Gamma` states) have degenerate components. It is both
possible to make single state calculations looking for the lowest
energy state of each symmetry or state-average calculations in each of
the symmetries. The identification of the |Dinfh| states can be
somewhat difficult. For instance, once we have computed one
|SAG| state it can be a :math:`^1\Gamma_g` or a :math:`^1\Sigma_g^+` state.
In this case the simplest solution is to compare the obtained
energy to that of the :math:`^1\Gamma_g` degenerate component in
:math:`B_{1g}` symmetry, which must be equal to the energy of the
:math:`^1\Gamma_g` state computed in :math:`A_g` symmetry. Other situations
can be more complicated and require a detailed analysis of the
wave function.
It is important to have clean d orbitals and the :kword:`SUPSym`
keyword may be needed to separate :math:`\delta` and :math:`\sigma`
(and :math:`\gamma` if g-type functions are used in the basis set)
orbitals in symmetry 1 (:math:`A_g`). The :kword:`AVERage` keyword
is not needed here because the :math:`\pi` and :math:`\phi` orbitals have
the same occupation for :math:`\Sigma` and :math:`\Gamma` states.
.. index::
single: Spin–orbit coupling
Finally, when states of different multiplicities are close in
energy, the spin--orbit coupling which mix the different states
should be included. The CASPT2 study of the :math:`\ce{Ni2}` molecule
in reference :cite:`Pou:94`, after considering all the mentioned
effects determined that the ground
state of the molecule is a :math:`0_g^+` state, a mixture of the
:math:`^1\Sigma_g^+` and :math:`^3\Sigma_g^-` electronic states.
For a review of the spin--orbit coupling and other important
coupling effects see reference :cite:`Peric:95`.
.. index::
single: High symmetry
single: Symmetry; High symmetry molecules
single: RASSCF; Supersymmetry
single: Option; Supersymmetry
single: RASSCF; Cleanup
single: Option; Cleanup
single: RASSCF; Average option
.. _TUT\:sec\:hsym:
High symmetry systems in |molcas|
---------------------------------
There are a large number of symmetry point groups in which |molcas|
cannot directly work. Although unusual in organic chemistry, some
of them can be easily found in inorganic compounds. Systems belonging
for instance to three-fold groups such as :math:`C_{3v}`, :math:`D_{3h}`, or :math:`D_{6h}`,
or to groups such :math:`O_h` or :math:`D_{4h}` must be computed using lower symmetry
point groups. The consequence is, as in linear molecules, that
orbitals and states belonging to different representations in the
actual groups, belong to the same representation in the lower symmetry
case, and *vice versa*. In the :program:`RASSCF` program it is
possible to prevent the orbital and configurational mixing caused by
the first situation. The :kword:`CLEAnup` and :kword:`SUPSymmetry`
keywords can be used in a careful, and somewhat tedious, way. The right
symmetry behaviour of the RASSCF wave function is then assured. It is
sometimes not a trivial task to identify the symmetry of the orbitals
in the higher symmetry representation and which coefficients must vanish.
In many situations the ground state wave function keeps the right
symmetry (at least within the printing accuracy) and helps to identify
the orbitals and coefficients. It is more frequent that the mixing
happens for excited states.
The reverse situation, that is, that orbitals (normally degenerated) which
belong to the same symmetry representation in the higher symmetry groups
belong to different representations in the lower symmetry groups cannot
be solved by the present implementation of the :program:`RASSCF` program.
The :kword:`AVERage` keyword, which performs this task in the linear molecules,
is not prepared to do the same in non-linear systems. Provided that the
symmetry problems mentioned in the previous paragraph are treated in
the proper way and the trial orbitals have the right symmetry, the :program:`RASSCF`
code behaves properly.
.. index::
single: Symmetry; Three-fold groups
single: Geometry
There is a important final precaution concerning the high symmetry systems:
the geometry of the molecule must be of the right symmetry. Any deviation
will cause severe mixings. :numref:`block:porph` contains the
:program:`SEWARD` input for the magnesium porphirin molecule. This is
a :math:`D_{4h}` system which must be computed :math:`D_{2h}` in |molcas|.
For instance, the :math:`x` and :math:`y` coordinates of atoms C1 and C5
are interchanged with equal values in :math:`D_{4h}` symmetry. Both
atoms must appear in the :program:`SEWARD` input because they
are not independent by symmetry in the :math:`D_{2h}` symmetry in which
|molcas| is going to work. Any deviation of the values,
for instance to put the :math:`y` coordinate to 0.681879 Å in C1
and the :math:`y` to 0.681816 Å in C5 and similar deviations for the other
coordinates, will lead to severe symmetry mixtures. This must be
taken into account when geometry data are obtained from other
program outputs or data bases.
.. index::
single: Porphyrine–Mg
.. extractcode-block:: none
:filename: advanced/SEWARD.Mg-Porphyrine.input
:caption: Sample input of the SEWARD program for the magnesium
porphirin molecule in the :math:`D_{2h}` symmetry}
:name: block:porph
&SEWARD &END
Title
Mg-Porphyrine D4h computed D2h
Symmetry
X Y Z
Basis set
C.ANO-S...3s2p1d.
C1 4.254984 .681879 .000000 Angstrom
C2 2.873412 1.101185 0.000000 Angstrom
C3 2.426979 2.426979 0.000000 Angstrom
C4 1.101185 2.873412 0.000000 Angstrom
C5 .681879 4.254984 0.000000 Angstrom
End of basis
Basis set
N.ANO-S...3s2p1d.
N1 2.061400 .000000 0.000000 Angstrom
N2 .000000 2.061400 0.000000 Angstrom
End of basis
Basis set
H.ANO-S...2s0p.
H1 5.109145 1.348335 0.000000 Angstrom
H3 3.195605 3.195605 0.000000 Angstrom
H5 1.348335 5.109145 0.000000 Angstrom
End of basis
Basis set
Mg.ANO-S...4s3p1d.
Mg .000000 .000000 0.000000 Angstrom
End of basis
End of Input
.. index::
single: Symmetry; Three-fold groups
single: Geometry
The situation can be more complex for some three-fold point groups
such as :math:`D_{3h}` or :math:`C_{3v}`. In these cases it is not possible
to input in the exact cartesian geometry, which depends on
trigonometric relations and relies on the numerical precision
of the coordinates entry. It is necessary then to use in the
:program:`SEWARD` input as much
precision as possible and check on the distance matrix of the
:program:`SEWARD` output if the symmetry of the system has been
kept at least within the output printing criteria.