This module offers access to the Constant-Temperature molecular dynamics defined by Nose-Hoover equations of motion (described in S.Nose JCP, 81 P511 (1984) and W.G. Hoover, Phy. Rev. A31, p1695 (1985)) This method has the advantage that it is a continuous dynamics with well defined conserved quantities. [Other temperature scaling methods, available in CHARMM (included Berendsen method in Leap-frog integrator) have discontinuous dynamics.]

The original Hamiltonian for Nose dynamics is defined as follow:

```
H = H0 + HB
= H0(p/s,q) + P^2/2Q + (f+1)kTlns (1)
```

where f is a degree of freedom of the physical system. This Hamiltonian was originally propoesed by Nose in his JCP paper. The equations of motions defined by Eq. (1) are solved numerically in order to achieve the canonical ensemble MD simulation. Hoovers extended the Nose’s analysis. He derives a slightly different set of equations of motions which dispense with the time-scaling parameter s by transforming p’ = p/s, S = lns and dt’ = dt/s. Then he define the conserved quantity:

`H = H0(p',q) + P^2/2Q + (f+1)kTS (2)`

This is not Hamiltonian any more, but this energy is conserved thourghout simulation.

In Nose-Hoover method, two different types of constant temperature methods can be called: (1) simple method: system coupled to one heat bath (2) multi-bath method: different parts of system can be coupled to different heat baths.

In addtion, Nose-Hoover method in this release can be invoked under parallel platforms. To run Nose-Hoover dynamics in parallel machines, this dynamics has to be called under a velocity verlet integrator. In order to call the velocity verlet, please refer to “dynamics.doc”.

Keywords in [...] are optional. Defaults values are given in (...).

To call Nose-Hoover MD method, “nose” must be specified in usual dynamics command, such as

```
DYNamics NOSE [RSTN] TREF (300.0) QREF (1.0D+50) NCYC (10) -
IUNO (-1) NSNOS (10) etc
```

```
NOSE :== specification of Nose-Hoover method with one heat bath
[RSTN]:== reset command for S in Eq (2)
TREF :== Desired Temperature of the system (Defult = 300K)
QREF :== magnitured of Coupling, Q, (Defult=1.0D+50 -> large QREF
leads the typical microcanonical ensembel MD simulation)
IUNO :== unit of file that output from Nose-Hoover can be
written. (describ more detail later section)
NSNOS :== Frequency of writing output from Nose-Hoover method
NCYC :== number of recursive loop of kinetic energy in order to
achieve the convergence values.
```

In order to call multi-heat bath method, you have to specify the following command before you call dynamics command.

```
NOSE I
CALL J atom-selection-option
COEF J QREF (1.0D+50) TREF (300)
TOL (1.0D-10)
NCYC (10)
[CLEAR]
END
DYNamics [RSTN] IUNO (-1) NSNOS (10) etc -
```

```
I :== number of heat bath in the system (Maxmum 10)
CALL :== assigns the specified atom into Jth heat bath
COEF :== assignments of Jth heat bath's coupling and
desired temperature (you can use the different
temperature in each heat bath.)
TOL :== tolerance of kinetic energy convergence
CLEAR :== clear all assignments in Nose-Hoover method
```

When entering the Nose-Hoover module for CHARMM, it is nessesary to specify the number of heat baths in the system. If you are going to use one heat bath, you can call Nose-Hoover module from DYNAmic module. But if you have a plan to use more than one, you have to specifiy it before calling DYNAmic module. Also Nose-Hoover methods are working with Verlet and Velocity Verlet algorithms in this points. If you want to used a velocity Verlet, specifiy it in DYNAmic module. (DYNAmic VVER .....). Otherwise, Verlet algorithm is used in the MD simulation. For parallel platforms, a velocity Verlet must be used in order to run Nose-Hoover dynamics.

QREF is the thermal inertia parameter which controls the rate of temperature fluctuations. Too high value of QREF results in slow enegy flow between the system and heat bath, and the limit of QREF –> infinity we regain conventional MD. On the other hand, if QREF is too small, long-lived, weakly damped oscillations of the energy occur, resulting in poor equilibration. It is necessary to choose right QREF to achieve satisfactory damping of these correlations. QREF should be between 10 and 1000.

Nose-Hoover equation of motion for scaling factor S shows that it depends on the kinetic energy of the system. In the same time, molecular velocity depends on scaling factor S. So in order to solve these equations of motion, it requires an iterative process to get convergence of kinetic energies. NCYC is a number of iteration of this process and TOL is a tolerance to reach convergence of kinetic energies. If SHAKE is used for molecules, NCYC is about 3 to 5 in order to reach convergence.

If more than one heat bath are used in the MD simulation, you have to specify which atoms belong to which heat bath. In order to assign them in particular heat bath, you have to use atom-selection module in CHARMM. See select.doc to use atom selection syntax.

Reset command to set S in Eq. (2) equl to zero. Reseting of scaling parameter S doesn’t change the dynamics of physical system at all. Sometimes S may become very large numbers during equilibration MD simulations, so setting S to zero may be necesrry

If you set IUNO a positive integer, the output of some quantities from Nose-Hoover method will be written in UNIT IUNO. This output contains following:

```
1st line: Number of step, Number of degree of freedom, NSNOS (defined
above)
```

Then at every NSNOS times, following output will be written:

```
Step, Time (ps), Total Energy
S value, Temperatue for each heat bath
```

By using this output file, you can calculate the average temperature of system in each heat bath. In addition to that, S value can be used in order to get free energy of the system [Check Branka and Parrinello, Mol Phys, 58, 989 (1986)]

One-heat-bath Selection

. . . SHAKE BOND ANGLE TOL 1.0D-9 UPDATE INBFRQ 5 IHBFRQ 0 - CUTIM 8.0 IMGFRQ 5 - ATOM SHIFT VATOM VSWITCH - CUTNB 8.0 CTOFNB 7.5 CTONNB 6.5 EPS 1.0 CDIE OPEN WRITE UNIT 31 CARD NAME @9FOR015.RST DYNA NOSE QREF 50.0 TREF 300.0 NCYC 5 FIRSTT 200 - STRT NSTEP 100 TIME 0.001 NPRINT 10 IPRFRQ 100 - IUNREA -30 IUNWRI 31 IUNCRD -1 IUNVEL -1 - KUNIT -41 NSAVC 50 NSAVV 50 ISVFRQ 1000

Two-heat-bath selection

In this example, ethane (ETHA) in water system is used.

NOSE 2 ! two heat baths are used. First all ! molecules are assigned into heat bath #1 CALL 2 SELE RESNAME ETHA END ! put ethane in heat bath #2 COEF 1 QREF 50.0 TREF 300.0 COEF 2 QREF 50.0 TREF 300.0 NCYC 5 END OPEN WRITE UNIT 31 CARD NAME @9FOR015.RST DYNA STRT NSTEP 100 TIME 0.001 NPRINT 10 IPRFRQ 100 - IUNREA -30 IUNWRI 31 IUNCRD -1 IUNVEL -1 FIRSTT 100.0 - KUNIT -41 NSAVC 50 NSAVV 50 ISVFRQ 1000