Skip to content

Operator File Interpretation

The generated operator file is a direct implementation of the vibronic Hamiltonian discussed in the previous sections. Every quantity appearing in the operator file corresponds to one of the mathematical terms in the Hamiltonian.

Rather than writing equations directly, MCTDH represents the Hamiltonian using predefined keywords and operators. Understanding the meaning of these keywords makes it much easier to inspect, modify, and debug operator files.

Overall Structure

The operator file is divided into three main sections.

Snippet 1 — Overall Structure

OP_DEFINE-SECTION

PARAMETER-SECTION

HAMILTONIAN-SECTION

end-operator

These sections respectively define

  • the calculation metadata,
  • all numerical parameters,
  • the complete vibronic Hamiltonian.

Vibrational Frequencies

The first parameters appearing in the parameter section are the harmonic vibrational frequencies.

Snippet 2 — Vibrational Frequencies

w1  = 0.01846982151, ev
w2  = 0.02008219442, ev
w3  = 0.02090629613, ev
...
w32 = 0.20286887049, ev

Each frequency corresponds to one selected normal mode.

Mathematically,

\[ w_i \equiv \omega_i \]

where

  • \(w_i\) is the parameter appearing in the operator file,
  • \(\omega_i\) is the harmonic vibrational frequency of the \(i^{\text{th}}\) active normal mode.

These frequencies define both the kinetic and harmonic potential energy operators.

Electronic Energies

The next quantities define the diabatic electronic energies.

Snippet 3 — Diagonal Electronic Energies

e11 = 0.00000000, ev
e22 = 3.75623767, ev
e33 = 4.47474718, ev
e44 = 4.35603436, ev

These correspond to

\[ E_1,\; E_2,\; E_3,\; E_4. \]

The first state is chosen as the reference, therefore

\[ E_1=0. \]

Interstate Electronic Couplings

Electronic coupling between different diabatic states is represented by

Snippet 4 — Interstate Couplings

e12 = 0.02338558
e13 = 0.46931326
e14 = -0.53188716
e23 = 0.09142652
e24 = -0.18522474
e34 = 0.17474197

These parameters correspond to

\[ V_{12}, V_{13}, V_{14}, V_{23}, V_{24}, V_{34}. \]

They determine the interaction between different electronic states.

Linear Vibronic Coupling

Linear coupling parameters appear as

Snippet 5 — Linear Coupling Constants

h11k1
h12k1
h13k1
h14k1

h22k1
h23k1
h24k1

h33k1
h34k1

h44k1

The notation can be interpreted as

Symbol Meaning
h Hamiltonian parameter
11 Electronic states involved
k Linear vibronic coupling
1 Vibrational mode number

For example,

h12k5

means

\[ k_{12}^{(5)}. \]

When used inside the Hamiltonian,

h12k5   |5 q

it represents

\[ k_{12}^{(5)}Q_5. \]

Quadratic Vibronic Coupling

Second-order coupling constants are represented by

Snippet 6 — Quadratic Coupling Constants

h11g1
h12g1
h13g1
h14g1

h22g1
h23g1
...

h44g1

The notation is identical except that

g

denotes a quadratic coupling constant.

Thus

h23g7

corresponds to

\[ g_{23}^{(7)}. \]

Inside the Hamiltonian,

0.5*h23g7   |7 q^2

represents

\[ \frac12 g_{23}^{(7)} Q_7^2. \]

Vibrational Operators

The Hamiltonian section introduces the vibrational operators.

Snippet 7 — Vibrational Operators

w1      |1 KE
0.5*w1  |1 q^2

These correspond to

Operator Mathematical Meaning
KE \( -\dfrac12\dfrac{\partial^2}{\partial Q^2}\)
q \(Q\)
q^2 \(Q^2\)

Thus,

w1 |1 KE

represents the kinetic energy operator,

\[ -\frac12 \frac{\partial^2}{\partial Q_1^2}, \]

while

0.5*w1 |1 q^2

represents

\[ \frac12 \omega_1Q_1^2. \]

Electronic Operators

The Hamiltonian also specifies the electronic states on which each term acts.

Instead of using a table, the correspondence is

  • S1&1 → \(|1\rangle\!\langle1|\)
  • S2&2 → \(|2\rangle\!\langle2|\)
  • S3&3 → \(|3\rangle\!\langle3|\)
  • S4&4 → \(|4\rangle\!\langle4|\)

These are diagonal projection operators.

Similarly,

  • S1&2 → \(|1\rangle\!\langle2|\)
  • S1&3 → \(|1\rangle\!\langle3|\)
  • S1&4 → \(|1\rangle\!\langle4|\)
  • S2&3 → \(|2\rangle\!\langle3|\)
  • S2&4 → \(|2\rangle\!\langle4|\)
  • S3&4 → \(|3\rangle\!\langle4|\)

represent interstate electronic operators.

Snippet 8 — Electronic Hamiltonian

e11     |el S1&1
e22     |el S2&2
e33     |el S3&3
e44     |el S4&4

e12     |el S1&2
e13     |el S1&3
e14     |el S1&4

Reading a Hamiltonian Line

Consider the line

Snippet 9

h12k7    |7 q    |el S1&2

This should be interpreted as

  • h12k7 → linear coupling constant,
  • |7 q → normal coordinate of mode 7,
  • |el S1&2 → interaction between electronic states 1 and 2.

The corresponding mathematical expression is

\[ k_{12}^{(7)} Q_7 |1\rangle\!\langle2|. \]

Similarly,

0.5*h22g3    |3 q^2    |el S2&2

represents

\[ \frac12 g_{22}^{(3)} Q_3^2 |2\rangle\!\langle2|. \]

Summary Table

Operator File Mathematical Quantity Physical Meaning
w1 \(\omega_1\) Vibrational frequency
KE \( -\frac12\dfrac{\partial^2}{\partial Q^2}\) Kinetic energy
q \(Q\) Normal coordinate
q^2 \(Q^2\) Harmonic coordinate
e11 \(E_1\) Diabatic energy
e12 \(V_{12}\) Electronic coupling
h12k1 \(k_{12}^{(1)}\) Linear coupling
h12g1 \(g_{12}^{(1)}\) Quadratic coupling
S1&2 $ 1\rangle!\langle2

Summary

The MCTDH operator file is a compact representation of the complete vibronic Hamiltonian. The Parameter Section defines the numerical quantities required by the calculation, while the Hamiltonian Section combines these parameters with vibrational and electronic operators to construct the final Hamiltonian. Understanding the correspondence between the operator syntax and the mathematical expressions enables users to interpret, validate, and customize operator files for a wide range of quantum dynamics calculations.