So-called “quasi-classical”
trajectories^{Karplus:1965, Porter:1974, Porter:1975} (QCT) put vibrational
energy into each mode in the initial velocity setup step, which can improve on
the results of purely classical simulations, for example in the calculation of
photoelectron^{Lambrecht:2011} or infrared
spectra.^{Ramos-Cordoba:2011} Improvements include better agreement of
spectral linewidths with experiment at lower temperatures and better agreement
of vibrational frequencies with anharmonic calculations.

The improvements at low temperatures can be understood by recalling that even at low temperature there is nuclear motion due to zero-point motion. This is included in the quasi-classical initial velocities, thus leading to finite peak widths even at low temperatures. In contrast to that the classical simulations yield zero peak width in the low temperature limit, because the thermal kinetic energy goes to zero as temperature decreases. Likewise, even at room temperature the quantum vibrational energy for high-frequency modes is often significantly larger than the classical kinetic energy. QCT-MD therefore typically samples regions of the potential energy surface that are higher in energy and thus more anharmonic than the low-energy regions accessible to classical simulations. These two effects can lead to improved peak widths as well as a more realistic sampling of the anharmonic parts of the potential energy surface. However, the QCT-MD method also has important limitations which are described below and that the user has to monitor for carefully.

In our QCT-MD implementation the initial vibrational quantum numbers are generated as random numbers sampled from a vibrational Boltzmann distribution at the desired simulation temperature. In order to enable reproducibility of the results, each trajectory (and thus its set of vibrational quantum numbers) is denoted by a unique number using the AIMD_QCT_WHICH_TRAJECTORY variable. In order to loop over different initial conditions, run trajectories with different choices for AIMD_QCT_WHICH_TRAJECTORY. It is also possible to assign initial velocities corresponding to an average over a certain number of trajectories by choosing a negative value. Further technical details of our QCT-MD implementation are described in detail in Appendix A of Ref. Lambrecht:2011.

AIMD_QCT_WHICH_TRAJECTORY

Picks a set of vibrational quantum numbers from a random distribution.

TYPE:

INTEGER

DEFAULT:

1

OPTIONS:

$n$
Picks the $n$th set of random initial velocities.
$-n$
Uses an average over $n$ random initial velocities.

RECOMMENDATION:

Pick a positive number if you want the initial velocities to correspond
to a particular set of vibrational occupation numbers and choose a
different number for each of your trajectories. If initial velocities
are desired that corresponds to an average over $n$ trajectories, pick a
negative number.

Below is a simple example input for running a QCT-MD simulation of the vibrational spectrum of water. Most input variables are the same as for classical MD as described above. The use of quasi-classical initial conditions is triggered by setting the AIMD_INIT_VELOC variable to QUASICLASSICAL.

$molecule 0 1 O 0.000000 0.000000 0.520401 H -1.475015 0.000000 -0.557186 H 1.475015 0.000000 -0.557186 $end $rem JOBTYPE freq INPUT_BOHR true METHOD hf BASIS 3-21g $end @@@ $molecule read $end $rem JOBTYPE aimd INPUT_BOHR true METHOD hf BASIS 3-21g SCF_CONVERGENCE 6 TIME_STEP 20 ! (in atomic units) AIMD_STEPS 1250 ! 600 fs total simulation time AIMD_TEMP 12 AIMD_PRINT 2 FOCK_EXTRAP_ORDER 6 ! Use a 6th-order extrapolation FOCK_EXTRAP_POINTS 12 ! of the previous 12 Fock matrices ! IR SPECTRAL SAMPLING AIMD_MOMENTS 1 AIMD_NUCL_SAMPLE_RATE 5 AIMD_NUCL_VACF_POINTS 1000 ! QCT-SPECIFIC SETTINGS AIMD_INIT_VELOC quasiclassical AIMD_QCT_WHICH_TRAJECTORY 1 ! Loop over several values to get ! the correct Boltzmann distribution. $end $end

Other types of spectra can be calculated by calculating spectral properties
along the trajectories. For example, we observed that photoelectron spectra can
be approximated quite well by calculating vertical detachment energies (VDEs)
along the trajectories and generating the spectrum as a histogram of the
VDEs.^{Lambrecht:2011} We have included several simple scripts in the
$QC/aimdman/tools subdirectory that we hope the user will find helpful
and that may serve as the basis for developing more sophisticated tools. For
example, we include scripts that allow to perform calculations along a
trajectory (`md_calculate_along_trajectory`

) or to calculate vertical
detachment energies along a trajectory (`calculate_rel_energies`

).

Another application of the QCT code is to generate random geometries sampled from the vibrational wave function via a Monte Carlo algorithm. This is triggered by setting both the AIMD_QCT_INITPOS and AIMD_QCT_WHICH_TRAJECTORY variables to negative numbers, say $-m$ and $-n$, and setting AIMD_STEPS to zero. This will generate $m$ random geometries sampled from the vibrational wave function corresponding to an average over $n$ trajectories at the user-specified simulation temperature.

AIMD_QCT_INITPOS

Chooses the initial geometry in a QCT-MD simulation.

TYPE:

INTEGER

DEFAULT:

0

OPTIONS:

$0$
Use the equilibrium geometry.
$n$
Picks a random geometry according to the harmonic vibrational wave function.
$-n$
Generates $n$ random geometries sampled from
the harmonic vibrational wave function.

RECOMMENDATION:

None.

For systems that are described well within the harmonic oscillator model and
for properties that rely mainly on the ground-state dynamics, this simple MC
approach may yield qualitatively correct spectra. In fact, one may argue that
it is preferable over QCT-MD for describing vibrational effects at very low
temperatures, since the geometries are sampled from a true quantum distribution
(as opposed to classical and quasi-classical MD). We have included another
script in the *$QC/aimdman/tools* directory to help with the calculation
of vibrationally averaged properties (`monte_geom`

).

$comment Generates 1000 random geometries for HCl based on the harmonic vibrational wave function at 1 Kelvin. The wave function is averaged over 1000 sets of random vibrational quantum numbers (\ie{}, the ground state in this case due to the low temperature). $end $molecule 0 1 H 0.000000 0.000000 -1.216166 Cl 0.000000 0.000000 0.071539 $end $rem JOBTYPE freq METHOD b3lyp BASIS 6-311++G** $end @@@ $molecule read $end $rem JOBTYPE aimd METHOD B3LYP BASIS 6-311++G** SCF_CONVERGENCE 1 MAX_SCF_CYCLES 0 TIME_STEP 20 (in atomic units) AIMD_STEPS 0 AIMD_INIT_VELOC quasiclassical AIMD_QCT_VIBSEED 1 AIMD_QCT_VELSEED 2 AIMD_TEMP 1 (in Kelvin) ! set aimd_qct_which_trajectory to the desired ! trajectory number AIMD_QCT_WHICH_TRAJECTORY -1000 AIMD_QCT_INITPOS -1000 $end

It is also possible make some modes inactive, *i.e.*, to put vibrational energy
into a subset of modes (all other are set to zero). The list of active modes
can be specified using the *$qct_active_modes* section. Furthermore, the
vibrational quantum numbers for each mode can be specified explicitly using the
*$qct_vib_distribution* input section. It is also possible to set the phases
using *$qct_vib_phase* (allowed values are 1 and $-$1). Below is a simple sample
input:

$qct_active_modes 1 $end $qct_vib_distribution 0 $end $qct_vib_phase 1 $end ...

Finally we turn to a brief description of the limitations of QCT-MD. Perhaps
the most severe limitation stems from the so-called “kinetic energy spilling”
problem,^{Czako:2010} which means that there can be an artificial transfer
of kinetic energy between modes. This can happen because the initial
velocities are chosen according to quantum energy levels, which are usually
much higher than those of the corresponding classical systems. Furthermore, the
classical equations of motion also allow for the transfer of non-integer
multiples of the zero-point energy between the modes, which leads to different
selection rules for the transfer of kinetic energy. Typically, energy spills
from high-energy into low-energy modes, leading to spurious “hot” dynamics.
A second problem is that QCT-MD is actually based on classical Newtonian
dynamics, which means that the probability distribution at low temperatures can
be qualitatively wrong compared to the true quantum distribution.^{Lambrecht:2011}

Q-Chem implements a routine to monitor the kinetic energy within each normal mode along the trajectory and that is automatically switched on for quasi-classical simulations. It is thus possible to monitor for trajectories in which the kinetic energy in one or more modes becomes significantly larger than the initial energy. Such trajectories should be discarded. (Alternatively, see Ref. Czako:2010 for a different approach to the zero-point leakage problem.) Furthermore, this monitoring routine prints the squares of the (harmonic) vibrational wave function along the trajectory. This makes it possible to weight low-temperature results with the harmonic quantum distribution to alleviate the failure of classical dynamics for low temperatures.