Natural Orbitals¶

In the previous chapter, we introduced the one-particle density matrix and saw that it contains information about the electronic distribution in a molecule.

However, the density matrix itself is rarely inspected directly.

Instead, it is diagonalized to obtain a new set of orbitals called natural orbitals, together with their corresponding natural occupation numbers.

Natural orbitals provide one of the clearest pictures of electron correlation and are invaluable for evaluating whether a chosen active space is appropriate.

What Are Natural Orbitals?¶

Natural orbitals are molecular orbitals obtained by diagonalizing the one-particle density matrix.

Mathematically,

\[ \boxed{ \mathbf D\mathbf U = \mathbf U\mathbf n } \]

where

\(\mathbf D\) is the one-particle density matrix,
\(\mathbf U\) contains the natural orbitals,
\(\mathbf n\) is a diagonal matrix containing the natural occupation numbers.

After diagonalization,

\[ \mathbf n= \begin{pmatrix} n_1 & 0 & 0 & \cdots \\ 0 & n_2 & 0 & \cdots \\ 0 & 0 & n_3 & \cdots \\ \vdots & \vdots & \vdots & \ddots \end{pmatrix} \]

Each diagonal element \(n_i\) represents the occupation of one natural orbital.

Why Are Natural Orbitals Useful?¶

The original molecular orbitals are optimized during the CASSCF procedure, but they are not necessarily the most informative for analysing electron correlation.

Natural orbitals are arranged according to their occupation numbers, making them much easier to interpret.

Instead of examining hundreds of CI coefficients,

CI Expansion

↓

Thousands of Configurations

↓

Difficult to Interpret

we simply inspect

Natural Orbitals

↓

Occupation Numbers

↓

Electron Correlation

Occupation Numbers¶

Each natural orbital has an occupation number between 0 and 2.

For example,

Occupation	Interpretation
2.00	Fully occupied
1.98	Almost doubly occupied
1.50	Strongly correlated
1.00	Half occupied
0.50	Partially occupied
0.02	Nearly empty
0.00	Empty

Unlike RHF,

these values are not restricted to 0 or 2.

Fractional occupations arise naturally because the CASSCF wavefunction is a superposition of many electronic configurations.

Comparing RHF and CASSCF¶

For a closed-shell RHF calculation,

Orbital

1

Occupation

2.00

Orbital

2

Occupation

2.00

Orbital

3

Occupation

0.00

Orbital

4

Occupation

0.00

The occupations are integers because only one electronic configuration contributes.

In contrast,

a CASSCF calculation may produce

Orbital

1

Occupation

1.98

Orbital

2

Occupation

1.73

Orbital

3

Occupation

0.29

Orbital

4

Occupation

0.01

The fractional occupations reveal that electrons are shared among several configurations.

Interpreting Occupation Numbers¶

Natural occupation numbers provide direct insight into electron correlation.

Occupations close to 2¶

1.99

↓

Essentially Doubly Occupied

These orbitals behave like inactive (core) orbitals.

Occupations close to 0¶

0.01

↓

Essentially Empty

These orbitals behave like virtual orbitals.

Occupations near 1¶

1.00

↓

Strong Multiconfigurational Character

These orbitals are typically essential members of the active space.

Intermediate Occupations¶

Values such as

indicate partial occupation and varying degrees of electron correlation.

These orbitals deserve closer inspection when validating an active space.

Using Occupation Numbers to Evaluate an Active Space¶

Natural occupations are one of the best indicators of whether an active space is well chosen.

Suppose a CASSCF calculation gives

Orbital	Occupation
12	1.99
13	1.98
14	1.01
15	0.98
16	0.04
17	0.01

Orbitals 14 and 15 clearly participate in electron correlation and should remain in the active space.

Orbitals 12 and 13 behave almost like inactive orbitals.

Orbitals 16 and 17 behave almost like virtual orbitals.

When Should the Active Space Be Expanded?¶

Sometimes an orbital outside the active space develops a significant fractional occupation.

For example,

Orbital	Occupation
20	0.32

Although orbital 20 was originally treated as virtual,

its occupation suggests that it contributes to the electronic structure.

This is often a sign that the active space should be expanded.

When Can the Active Space Be Reduced?¶

Conversely,

suppose an active orbital has

Occupation

1.9998

or

Occupation

0.0003

throughout the calculation.

Such orbitals behave almost exactly like inactive or virtual orbitals.

Removing them may simplify the calculation without significantly affecting the results.

However, active-space reduction should always be guided by chemical intuition and the nature of the problem being studied.

Practical Guidelines¶

The following table provides a useful rule of thumb.

Occupation Number	Interpretation	Recommendation
> 1.98	Inactive-like	Usually remain outside the active space
1.80–1.98	Weak correlation	Inspect carefully
0.20–1.80	Strong correlation	Strong candidate for the active space
0.02–0.20	Weak virtual correlation	Inspect if chemically relevant
< 0.02	Virtual-like	Usually remain outside the active space

These are practical guidelines rather than strict rules.

The chemical problem being studied should always guide the final decision.

Relation to GAMESS¶

GAMESS can print natural orbitals after a successful CASSCF calculation.

These orbitals are particularly useful for

analysing electron correlation,
selecting active spaces for larger calculations,
preparing XMCQDPT calculations,
studying excited states,
understanding bonding changes along reaction coordinates.

Many users inspect the natural occupations before deciding whether to modify the active space.

Looking Ahead¶

At this stage, we have covered every major theoretical component of the CASSCF method:

molecular orbitals,
Configuration State Functions,
the multiconfigurational wavefunction,
the Hamiltonian,
Configuration Interaction,
orbital optimization,
state averaging,
density matrices,
natural orbitals.

The next chapter brings all of these concepts together into a single workflow, showing how a CASSCF calculation proceeds inside GAMESS from the initial RHF orbitals to the final converged wavefunction.

Key Takeaways¶

Natural orbitals are obtained by diagonalizing the one-particle density matrix.
Their occupation numbers range from 0 to 2.
Fractional occupations are direct evidence of electron correlation.
Orbitals with occupations near 1 are typically the most important members of the active space.
Natural occupations provide one of the best diagnostics for validating or refining an active space.
They are widely used before proceeding to methods such as XMCQDPT.