# Projection analysis¶

## Theory¶

A serious flaw of Mulliken population analysis is its basis set sensitivity. This is well known from the non-relativistic domain. In the relativistic domain a further disadvantage is that the expansion of large and small components in separate basis sets means large and small component populations come separately and it is not easy to see how they are connected. Furthermore, if the large and small component are expanded in scalar basis functions, as is the case of the DIRAC code, then the population analysis can not distinguish distributions from different spin-orbit components of atomic orbitals, for instance, \(p_{1/2}\) and \(p_{3/2}\).

Projection analysis [Dubillard2006] eliminates all these inconveniences. It is based on the simple concept of molecular orbitals as linear combination of atomic orbitals (LCAO) and allows analysis of electronic structure in terms of well-defined atomic (or fragment) orbitals.

Projection analysis requires as input a set of pre-calculated orbitals {\(\psi^A_p\)} for the constituent atoms (or fragment) of the molecule. Here \(A\) refers to the atomic center (or fragment) and \(p\) is on orbital index for that center. The molecular orbitals are then expanded as

Typically one will select the occupied orbitals of the constituent atoms. They are not
guaranteed to fully span the molecular orbitals and therefore the expansion includes
their orthogonal complement \(\left|\psi^{pol}_i\right>\), denoted the *polarization contribution*.
The expansion coefficients \(c^A_{pi}\) are found by solving the equation

obtained by projection.

For projection analysis, arbitrary fragments can in principle be used. There is, however, an important restriction
in that DIRAC can only do analysis in terms of fragments complying to the molecular point group. For instance,
the two hydrogens in water are related by symmetry in the full \(C_{2v}\) molecular point group. They then have
to be treated as a single fragment, or they can be separated by lowering the symmetry. Starting from DIRAC14
projection analysis can be carried out in terms of individual atoms using the *.ATOMS* keyword, as shown in
the example below.

## Example 1: Methane¶

### Introduction¶

As a first simple example we shall consider the methane molecule.
The molecular input file `CH4.xyz` (`download`) is

```
5
C 0.0000000000 0.0000000000 0.0000000000
H 0.6298891440 0.6298891440 -0.6298891440
H 0.6298891440 -0.6298891440 0.6298891440
H -0.6298891440 0.6298891440 0.6298891440
H -0.6298891440 -0.6298891440 -0.6298891440
```

Suppose that we have run a DFT(PBE) calculation using:

```
pam --inp=PBE --mol=CH4.xyz --outcmo
```

where we keep the coefficients and the menu file `PBE.inp` (`download`) is

```
**DIRAC
.WAVE FUNCTION
**HAMILTONIAN
.DFT
PBE
**WAVE FUNCTION
.SCF
*SCF
.CLOSED SHELL
10
**MOLECULE
*BASIS
.DEFAULT
cc-pVDZ
*END OF
```

### Preparing for analysis¶

We would now like to analyze the electronic structure of the methane molecule in terms of the orbitals of the constituent atoms. We first generate the atomic fragments. We recommend calculating the atoms in the highest possible symmetry and the converting the coefficients to no (\(C_1\)) symmetry. This allows a very precise identification of the individual atomic orbitals (including \(m_j\) quantum number). We accordingly run:

```
pam --inp=H --mol=H --get "DFACMO=ac.H"
pam --inp=C --mol=C --get "DFACMO=ac.C"
```

with corresponding input files `H.inp` (`download`),
`H.mol` (`download`),
`C.inp` (`download`),
and `C.mol` (`download`).

As a final preparation we make symbolic links to the needed coefficient files:

```
$ ln -s ac.C AFCXXX
$ ln -s ac.H AFHXXX
```

### Running the analysis¶

We are now ready for the projection analysis itself. We prepare the input file `prj.inp` (`download`)

```
**DIRAC
.ANALYZE
**HAMILTONIAN
.DFT
PBE
**WAVE FUNCTION
.SCF
*SCF
.CLOSED SHELL
10
**ANALYZE
.PROJEC
*PROJEC
#.POLREF
.ATOMS
AFCXXX
1..5
AFHXXX
1
**MOLECULE
*BASIS
.DEFAULT
cc-pVDZ
*END OF
```

In the `*PROJEC` section we specify the two atomic types identified by the name of
their coefficient files as well as an orbital string (see *Specification of orbital strings*) listing
what orbitals to include for each atom. DIRAC will assume that each atom has been calculated
in its proper basis.

We now run the projection analysis using:

```
pam --inp=prj --mol=CH4.xyz --incmo --copy="AF*"
```

### Interpreting the output¶

The first thing to look for in the output is the section:

```
* Total gross contributions:
AFCX 1 6.4301 E - 6.4301 P - 0.0000
AFHX 1 0.8716 E - 0.8716 P - 0.0000
AFHX 2 0.8716 E - 0.8716 P - 0.0000
AFHX 3 0.8716 E - 0.8716 P - 0.0000
AFHX 4 0.8716 E - 0.8716 P - 0.0000
Polarization: 0.0834
```

It shows the gross population (in Mulliken sense, but in terms of the fragment orbitals) of each atom
and then, most importantly, the gross population of the polarization contribution. In this example it
is small and perfectly acceptable. If it becomes too large the analysis looses meaning because it
impies that significant electron density has not been assigned to any fragment. In the case of
significant polarization contribution one can either increase the number of atomic orbitals used
in the analysis or turn on *repolarization*. For the latter it suffices to add the *.POLREF*
keyword which activates *Intrinsic Atomic Orbitals (IAOs)*, [Knizia2013],
thus eliminating completely the polarization contribution. We then get:

```
* Total gross contributions:
AFCX 1 6.4782 E - 6.4782 P - 0.0000
AFHX 1 0.8804 E - 0.8804 P - 0.0000
AFHX 2 0.8804 E - 0.8804 P - 0.0000
AFHX 3 0.8804 E - 0.8804 P - 0.0000
AFHX 4 0.8804 E - 0.8804 P - 0.0000
Polarization: 0.0000
```

from which we can conclude that at this level of theory the carbon and hydrogen charges in methane are -0.48e and +0.12e, respectively.

Having concluded that the polarization contribution is acceptable, we can proceed with the analysis. We turn to this section:

```
* Total reference orbital contributions:
AFCX 1 E 1 1.99999 -0.10660951E+02
AFCX 1 E 2 1.27874 -0.97828469E+00
AFCX 1 E 3 1.05077 -0.65580701E+00
AFCX 1 E 4 1.05030 -0.65541446E+00
AFCX 1 E 5 1.05030 -0.65541446E+00
AFHX 1 E 1 0.87162 -0.23118508E+00
AFHX 2 E 1 0.87162 -0.23118508E+00
AFHX 3 E 1 0.87162 -0.23118508E+00
AFHX 4 E 1 0.87162 -0.23118508E+00
```

which gives the accumulated gross population of each fragment orbital. We can identify
the fragments from the name of the coefficient file and its orbital energy. For instance
the first orbital on the list is clearly carbon *1s* followed by carbon *2s*. Next comes three
orbitals that are almost degenerate. These are cleary the carbon *2p* orbitals, the first one
being \(2p_{1/2}\) followed by the two \(2p_{3/2}\). By furthermore looking into the
output of the carbon atom calculation (note that we asked for Mulliken population analysis in the
input) we find that the first \(2p_{3/2}\) orbital has \(m_j=1/2\) and the
second \(m_j=-3/2\). From these gross population we can deduce the atomic configurations of
the atoms in the molecule. For carbon we find \([He]2s^{1.3}2p^{3.2}\) and for hydrogen
\(1s^{0.9}\).

There is also detailed output for each molecular orbital in the output. For the third molecular orbital we find for instance:

```
* Electronic eigenvalue nr. 3: -0.3411330794649 (Occupation : f = 1.0000)
================================================
Orbital Total Eigenvalue Kramers partner 1 Kramers partner 2
AFCX 1 E 3 0.57945 -0.65580701E+00 (-1.0000,-0.0000) ( 0.0000, 0.0000)
AFHX 1 E 1 0.31559 -0.23118508E+00 ( 0.0000,-0.5774) (-0.5774,-0.5774)
AFHX 2 E 1 0.31559 -0.23118508E+00 ( 0.0000, 0.5774) ( 0.5774,-0.5774)
AFHX 3 E 1 0.31559 -0.23118508E+00 ( 0.0000, 0.5774) (-0.5774, 0.5774)
AFHX 4 E 1 0.31559 -0.23118508E+00 ( 0.0000,-0.5774) ( 0.5774, 0.5774)
* Gross contributions:
AFCX 1 0.5254 E - 0.5254 P - 0.0000
AFHX 1 0.1165 E - 0.1165 P - 0.0000
AFHX 2 0.1165 E - 0.1165 P - 0.0000
AFHX 3 0.1165 E - 0.1165 P - 0.0000
AFHX 4 0.1165 E - 0.1165 P - 0.0000
Polarization: 0.0085
```

To fully understand the output we have to keep in mind that all calculations are Kramers-restricted such that all orbitals come in pairs. The expansion of the molecular orbitals should therefore rather be written as

where \(c^A_{pi}\) and \(c^A_{\overline{p}i}\) is the expansion coefficient of Kramers partner 1 and 2, respectively. Let us now define the absolute value

This is the real number reported under the heading `Total`. Under the heading
`Kramers partner 1` is reported the complex number

and analogously for Kramers partner 2, from which we get the detailed decomposition of the molecular orbital. We find that this molecular orbital is composed of the carbon \(2p_{1/2}\) orbital and the hydrogen \(1s\) orbitals.

### Exercises¶

- What is the effect of increasing the C-H bond distance on the atomic configurations of the C,H atoms in the methane molecule and on the atomic charges of the constituent atoms ? Collect calculated data in a table.
- How different basis sets affect the atomic configurations of the C,H atoms in the methane molecule and the atomic charges of the same constituent atoms ?
- Investigate the effect of various DFT functionals (see
**DFT*) on the atomic configurations of the C and H atoms in the methane molecule and on atomic charges of these constituent atoms.

## Example 2: Uranyl¶

### Introduction¶

We next turn to a molecule with a heavy atom and a more complicated electronic structure, namely uranyl
\(UO_2^{2+}\). Here the two oxygens are related by symmetry, and so one would at first consider
running the projection analysis with lower (or no) symmetry or using the *.ATOMS* keyword, as
in the methane example above. However, we can reduce the symmetry from \(D_{\infty h}\) to \(C_{\infty v}\)
by introducing a ghost center. Our molecular input file then looks like

```
DIRAC
uranyl
....
C 3 A
92. 1
U 0.0 0.0 0.00000
LARGE BASIS dyall.v3z
8. 2
O1 0.0 0.0 1.70447861
O2 0.0 0.0 -1.70447861
LARGE BASIS cc-pVTZ
0. 1
Gh 0.0 0.0 10.0
LARGE 0
FINISH
```

Now the oxygen atoms are no longer connected by symmetry and we can carry out the projection analysis in \(C_{\infty v}\) symmetry.

### Preparing the analysis¶

Using the menu file `HF.inp`

```
**DIRAC
.ANALYZE
.WAVE FUNCTION
**INTEGRALS
*READIN
.UNCONT
**WAVE FUNCTION
.SCF
*SCF
.CLOSED SHELL
106
**END OF
```

we run 4-component Hartree-Fock:

`pam --mw=120 --inp=HF --mol=UO2 --get "DFCOEF=cf.UO2"`

saving the coefficients. Next we generate atomic reference orbitals. For oxygen this is quite straightforward:

`pam --inp=O --mol=O --get "DFCOEF=cf.O"`

using molecule input `O.mol`

```
DIRAC
UO2
cc-pVTZ basis
C 2 A
8. 1
O 0.0 0.0 0.0
LARGE BASIS cc-pVTZ
0. 1
Gh 0.0 0.0 10.0
LARGE 0
FINISH
```

and the menu file `O.inp`

```
**DIRAC
.ANALYZE
.WAVE FUNCTION
**INTEGRALS
*READIN
.UNCONT
**WAVE FUNCTION
.SCF
*SCF
.CLOSED SHELL
4
.OPEN SHELL
1
4/6
**ANALYZE
.MULPOP
**END OF
```

For the uranium we must be a bit more careful since the ground state configuration is an open-shell one: \([Rn]5f^36d^17s^2\).
Using the molecular input `U.mol`

```
DIRAC
UO2
cc-pVTZ basis
C 2 A
92. 1
U 0.0 0.0 0.00000
LARGE BASIS dyall.v3z
0. 1
Gh 0.0 0.0 10.0
LARGE 0
FINISH
```

we first calculate the \(U^{3+}\) cation with configuration \([Rn]5f^3\)

```
pam --inp=UIII --mol=U --outcmo
```

which converges easily using the menu file `UIII.inp`

```
**DIRAC
.ANALYZE
.WAVE FUNCTION
**INTEGRALS
*READIN
.UNCONT
**WAVE FUNCTION
.SCF
*SCF
.CLOSED SHELL
86
.OPEN SHELL
1
3/14
**ANALYZE
.MULPOP
*MULPOP
.VECPOP
1..oo
**END OF
```

Note that we ask for a Mulliken population analysis of all positive-energy orbitals. This is in order to find the *6d* and *7s* orbitals.
Once identified, we bring them into correct position using reordering and keep them there using static overlap selection as shown in
the menu file `U.inp`

```
**DIRAC
.ANALYZE
.WAVE FUNCTION
**INTEGRALS
*READIN
.UNCONT
**WAVE FUNCTION
.REORDER
1..43,51,44..50,52..56
.SCF
*SCF
.CLOSED SHELL
88
.OPEN SHELL
2
3/14
1/10
.OVLSEL
.NODYNSEL
**ANALYZE
.MULPOP
*MULPOP
.VECPOP
1..oo
**END OF
```

Convergence is now straightforward

`pam --incmo --inp=U --mol=U --get "DFCOEF=cf.U"`

and the correct configuration is confirmed by Mulliken population analysis.

### Running the analysis¶

We are now ready to do the projection analysis. We prepare the input file `prj.inp`

```
**DIRAC
.ANALYZE
**INTEGRALS
*READIN
.UNCONT
**WAVE FUNCTION
.SCF
*SCF
.CLOSED SHELL
106
**ANALYZE
.PROJEC
*PROJEC
.OWNBAS
.VECREF
3
AFUXXX
1
1..56
AFO1XX
1
1..5
AFO2XX
1
1..5
**END OF
```

and prepare the coefficient files:

```
$ ln -s cf.O AFO1XX
$ ln -s cf.O AFO2XX
$ ln -s cf.U AFUXXX
$ cp cf.UO2 DFCOEF
```

We run the projection analysis:

```
pam --incmo --inp=prj --mol=UO2 --copy="AF*" --mw=120
```

### Interpreting the output¶

As in the methane case above we first look at the polarization contribution:

```
* Total gross contributions:
AFUXXX 89.0635 E - 89.0635 P - 0.0000
AFO1XX 8.3098 E - 8.3098 P - 0.0000
AFO2XX 8.3098 E - 8.3098 P - 0.0000
Polarization: 0.3169
```

We see that it is more sizable than before. If we eliminate the polarization contribution using the *.POLREF* keyword, we obtain:

```
* Total gross contributions:
AFUXXX 89.1586 E - 89.1586 P - 0.0000
AFO1XX 8.4207 E - 8.4207 P - 0.0000
AFO2XX 8.4207 E - 8.4207 P - 0.0000
Polarization: 0.0000
```

The atomic gross populations tell us that the oxygen atoms has a charge -0.42, whereas the uranium atom has charge +2.84, considerably far from the formal oxidation state +VI.

We next turn to the electron configuration of the atoms in the molecule by looking at the section:

```
* Total reference orbital contributions:
AFUXXX E 1 2.00000 -0.42818128E+04 1s
AFUXXX E 2 2.00000 -0.80663682E+03 2s
AFUXXX E 3 2.00000 -0.77703498E+03 2p-
AFUXXX E 4 2.00000 -0.63578312E+03 2p+
AFUXXX E 5 2.00000 -0.63578312E+03 2p+
AFUXXX E 6 2.00000 -0.20673001E+03 3s
AFUXXX E 7 2.00000 -0.19325143E+03 3p-
AFUXXX E 8 2.00000 -0.16037784E+03 3p+
AFUXXX E 9 2.00000 -0.16037784E+03 3p+
AFUXXX E 10 2.00000 -0.13906994E+03 3d-
AFUXXX E 11 2.00000 -0.13906994E+03 3d-
AFUXXX E 12 2.00000 -0.13242590E+03 3d+
AFUXXX E 13 2.00000 -0.13242590E+03 3d+
AFUXXX E 14 2.00000 -0.13242590E+03 3d+
AFUXXX E 15 2.00000 -0.54355461E+02 4s
AFUXXX E 16 2.00000 -0.48231971E+02 4p-
AFUXXX E 17 2.00000 -0.39554187E+02 4p+
AFUXXX E 18 2.00000 -0.39554187E+02 4p+
AFUXXX E 19 2.00000 -0.29743891E+02 4d-
AFUXXX E 20 2.00000 -0.29743891E+02 4d-
AFUXXX E 21 2.00000 -0.28130201E+02 4d+
AFUXXX E 22 2.00000 -0.28130201E+02 4d+
AFUXXX E 23 2.00000 -0.28130201E+02 4d+
AFUXXX E 24 2.00000 -0.15202067E+02 4f-
AFUXXX E 25 2.00000 -0.15202067E+02 4f-
AFUXXX E 26 2.00000 -0.15202067E+02 4f-
AFUXXX E 27 2.00000 -0.14785575E+02 4f+
AFUXXX E 28 2.00000 -0.14785575E+02 4f+
AFUXXX E 29 2.00000 -0.14785575E+02 4f+
AFUXXX E 30 2.00000 -0.14785575E+02 4f+
AFUXXX E 31 2.00001 -0.12603340E+02 5s
AFUXXX E 32 1.99989 -0.10135890E+02 5p-
AFUXXX E 33 1.99952 -0.80948982E+01 5p+
AFUXXX E 34 1.99993 -0.80948982E+01 5p+
AFUXXX E 35 1.99983 -0.43524477E+01 5d-
AFUXXX E 36 1.99914 -0.43524477E+01 5d-
AFUXXX E 37 1.99872 -0.40407150E+01 5d+
AFUXXX E 38 1.99956 -0.40407150E+01 5d+
AFUXXX E 39 1.99990 -0.40407150E+01 5d+
AFUXXX E 40 1.98417 -0.21392050E+01 6s
AFUXXX E 41 1.93819 -0.13442976E+01 6p-
AFUXXX E 42 1.75124 -0.98481977E+00 6p+ (1/2)
AFUXXX E 43 1.98199 -0.98481977E+00 6p+ (3/2)
AFUXXX E 44 0.03906 -0.20241412E+00 7s
AFUXXX E 45 0.82745 -0.34596371E+00 5f- (1/2)
AFUXXX E 46 0.15661 -0.34596371E+00 5f- (3/2)
AFUXXX E 47 0.00000 -0.34596371E+00 5f- (5/2)
AFUXXX E 48 0.00000 -0.31822326E+00 5f+ (7/2)
AFUXXX E 49 0.00000 -0.31822326E+00 5f+ (5/2)
AFUXXX E 50 0.33927 -0.31822326E+00 5f+ (3/2)
AFUXXX E 51 0.94161 -0.31822326E+00 5f+ (1/2)
AFUXXX E 52 0.42621 -0.19266767E+00 6d- (1/2)
AFUXXX E 53 0.09667 -0.19266767E+00 6d- (3/2)
AFUXXX E 54 0.34518 -0.18309790E+00 6d+ (1/2)
AFUXXX E 55 0.33432 -0.18309790E+00 6d+ (3/2)
AFUXXX E 56 0.00010 -0.18309790E+00 6d+ (5/2)
AFO1XX E 1 2.00005 -0.20696087E+02 1s
AFO1XX E 2 1.89709 -0.12503665E+01 2s
AFO1XX E 3 1.51690 -0.61398417E+00 2p-
AFO1XX E 4 1.46077 -0.61259798E+00 2p+
AFO1XX E 5 1.54590 -0.61259798E+00 2p+
AFO2XX E 1 2.00005 -0.20696087E+02 1s
AFO2XX E 2 1.89709 -0.12503665E+01 2s
AFO2XX E 3 1.51690 -0.61398417E+00 2p-
AFO2XX E 4 1.46077 -0.61259798E+00 2p+
AFO2XX E 5 1.54590 -0.61259798E+00 2p+
```

We can to a large extent deduce the identity of the various atomic orbitals by just looking at orbital energies
and I have added it by hand to the above output in a hopefully obvious notation. Even more precise information,
such as \(m_j\) value (indicated in parenthesis) can be obtained from looking at the Mulliken population analysis of each atom.
From this list of gross population we extract the atomic valence configuration \(5f^{2.25}6d^{1.16}7s^{0.04}\) for
the uranium atom and \(2s^{1.90}2p^{4.41}\) for oxygen. An interesting feature is that the uranium *6p* population
adds up to *5.64* and not six electrons. This is a manifestation of the so-called *6p-hole*, due to overlap with the oxygen ligands,
and mostly located to the \(6p_{3/2.1/2}\) orbital.