Operator File Interpretation¶

The previous chapters introduced the mathematical form of the vibronic Hamiltonian and showed how the vibrational, electronic, linear, and quadratic coupling terms are combined to describe the coupled electron-nuclear dynamics.

In practice, however, MCTDH does not read these equations directly. Instead, the Hamiltonian is written using the operator-file syntax, where every keyword represents a mathematical operator or parameter. Understanding this correspondence makes it much easier to interpret, modify, and debug operator files.

This section summarizes the relationship between the mathematical Hamiltonian and the generated MCTDH operator file.

Overall Structure¶

Every generated operator file consists of three main parts.

OP_DEFINE-SECTION
        │
        ▼
PARAMETER-SECTION
        │
        ▼
HAMILTONIAN-SECTION

The first section provides descriptive information about the calculation, the second defines all numerical parameters, and the third constructs the Hamiltonian using those parameters.

Vibrational Parameters¶

The vibrational frequencies appear in the parameter section as

w1
w2
w3
...

where

\[ w_i \equiv \omega_i \]

is the harmonic frequency of the \(i^{\text{th}}\) active normal mode.

These frequencies are obtained from the harmonic vibrational analysis and determine the kinetic and potential energy operators for each mode.

Electronic Parameters¶

The diagonal electronic energies are written as

e11
e22
e33

representing

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

The off-diagonal parameters

e12
e13
e23

represent the electronic coupling matrix elements

\[ V_{12},\; V_{13},\; V_{23}. \]

These quantities are extracted directly from the diabatic electronic structure calculations.

Linear Coupling Parameters¶

Linear vibronic coupling constants are represented by

h12_k5

This notation can be interpreted as

Symbol	Meaning
`h`	Hamiltonian parameter
`12`	Coupling between electronic states 1 and 2
`k`	Linear vibronic coupling
`5`	Vibrational mode 5

Mathematically,

\[ h12\_k5 \equiv k_{12}^{(5)}. \]

When used inside the Hamiltonian,

h12_k5    |5 q

it represents

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

Quadratic Coupling Parameters¶

Quadratic coupling parameters use the notation

h12_g5

where

Symbol	Meaning
`h`	Hamiltonian parameter
`12`	Coupled electronic states
`g`	Quadratic vibronic coupling
`5`	Vibrational mode

which corresponds to

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

Within the Hamiltonian,

0.5*h12_g5    |5 q^2

represents

\[ \frac12 g_{12}^{(5)} Q_5^2. \]

The factor of one-half arises from the second-order Taylor expansion of the electronic Hamiltonian.

Coordinate Operators¶

The Hamiltonian section contains several operator keywords.

Operator	Mathematical Meaning
`KE`	\( -\frac12\dfrac{\partial^2}{\partial Q^2} \)
`q`	\(Q\)
`q^2`	\(Q^2\)

Thus,

w5    |5 KE

represents

\[ -\frac12 \frac{\partial^2}{\partial Q_5^2}, \]

while

0.5*w5    |5 q^2

represents

\[ \frac12 \omega_5Q_5^2. \]

Electronic Operators¶

Electronic operators specify which electronic states are connected by a Hamiltonian term.

Examples include

S1&1
S2&2
S1&2
S2&3

Their mathematical interpretation is

S1&1 → \(|1\rangle\!\langle1|\)
S2&2 → \(|2\rangle\!\langle2|\)
S1&2 → \(|1\rangle\!\langle2|\)
S2&3 → \(|2\rangle\!\langle3|\)

Diagonal operators contribute to the electronic energies, while off-diagonal operators describe transitions between electronic states.

Reading a Hamiltonian Line¶

Consider the operator-file entry

h12_k7    |7 q    |el S1&2

This line can be interpreted as

h12_k7 — 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*h22_g3    |3 q^2    |el S2&2

represents

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

Complete Mapping¶

The table below summarizes the correspondence between the operator-file syntax and the mathematical Hamiltonian.

Operator File	Mathematical Quantity	Physical Meaning
`w1`	\(\omega_1\)	Vibrational frequency
`KE`	\( -\frac12\partial^2/\partial Q^2 \)	Kinetic energy
`q`	\(Q\)	Normal coordinate
`q^2`	\(Q^2\)	Harmonic coordinate
`e11`	\(E_1\)	Diabatic state energy
`e12`	\(V_{12}\)	Electronic coupling
`h11_k1`	\(k_{11}^{(1)}\)	Linear intrastate coupling
`h12_k1`	\(k_{12}^{(1)}\)	Linear interstate coupling
`h11_g1`	\(g_{11}^{(1)}\)	Quadratic intrastate coupling
`h12_g1`	\(g_{12}^{(1)}\)	Quadratic interstate coupling
`S1&2`	(	1\rangle\langle2

Summary¶

Every line in an MCTDH operator file represents a specific term in the vibronic Hamiltonian. The parameter section defines the numerical values of the vibrational frequencies, electronic energies, and vibronic coupling constants, while the Hamiltonian section combines these quantities with the vibrational and electronic operators to construct the complete Hamiltonian. Understanding this correspondence provides a direct link between the mathematical formulation of the vibronic Hamiltonian and its implementation within MCTDH, making it easier to interpret, validate, and modify operator files for quantum dynamics calculations.