What is the physical origin of the difference between AUXILIARY and HESSIAN matrices?

[jgSi1113]

Dear all,

I performed a NPT calculation via SSCHA+QE and the minimization created a dynamical matrices, which should be the auxiliary SSCHA dynamical matrices used in arXiv:2411.03822v1. But very recently, I found there is another paper used the Hessian matrix to calculate the phonon dispersion. The later paper says "8 materials exhibit stable auxiliary SSCHA dynamical matrices, but their Hessian dynamical matrices still show imaginary frequencies..". What is the physical difference between the them? Which one is the more meaningful for the SSCHA calculations? [Although I think the later one is better.]

I have finshed the minimization and obtained the auxiliary SSCHA dynamical matrices, How can I transfer them into the Hessian ones?

Any suggestions and discussions will be greatly appreciated.

Thanks and Regards!
Jianguo Si

[DjordjeDangic]

To have a complete understanding of what are auxiliary and Hessian phonons please read these papers in detail: https://iopscience.iop.org/article/10.1088/1361-648X/ac066b/meta and https://journals.aps.org/prb/abstract/10.1103/PhysRevB.96.014111 .

However, in short, you can think of auxiliary phonons as the first correction to the harmonic phonons, while Hessian would be a second-order correction. One problem is that auxiliary phonons are always positive so they can not tell you anything about the dynamical stability of the system. If you want to see whether your system is dynamically stable (which is usually why one calculates phonons) you need to calculate free energy Hessian phonons.

[jgSi1113]

Thanks for your kindly reply. It is very nice for me to understand the difference between auxiliary and Hessian.

Best & Regards!
Jianguo

[mesonepigreco]

To compute the Hessian phonons, there is a simple function

hessian = ensemble.ensemble.get_free_energy_hessian(include_v4=False)
hessian.save_qe("hessian")

As Djordje said, the Hessian dispersion is essential for determining the stability of a structure. If you need the phonons measured from X-Ray diffraction, you should compute the full spectral function. A review of the differences between all kinds of phonons in an anharmonic system is arxiv:2407.03090.

We have two nice tutorials for both of them, where also some fundamental aspects, like your questions, are discussed:

• Hessian Tutorial
• Spectral function Tutorial

[jgSi1113]

Thanks for your kindly reply.

I heve do similar work to obtain the Hessian matrix [ But to be honest, it needs a lot of time.].

Please kindly mark this topic as Answered.

Best & Regards!
Jianguo

[jgSi1113]

To compute the Hessian phonons, there is a simple function

hessian = ensemble.ensemble.get_free_energy_hessian(include_v4=False)
hessian.save_qe("hessian")

As Djordje said, the Hessian dispersion is essential for determining the stability of a structure. If you need the phonons measured from X-Ray diffraction, you should compute the full spectral function. A review of the differences between all kinds of phonons in an anharmonic system is arxiv:2407.03090.

We have two nice tutorials for both of them, where also some fundamental aspects, like your questions, are discussed:

• Hessian Tutorial
• Spectral function Tutorial

Hi, Could you like to help me to evaluate the time I needed to calculated the Hessian with include_v4=True? If I set the include_v4= False, the calculation only needs several minutes. But if I set the include_v4=True, the calculation need some days. For me, the several final lines in the log file says:

=== DEBUG ODD STRAIGHT ===
N_MODE: 336
NTYP: 3
NAT_SC: 112
=== DEBUG GET_CMAT ===
N_MODE: 336
NAT_SC: 112
NTYP: 3
CMAT: ALLOCATION
CMAT: BEFORE EMAT
=== DEBUG GET_EMAT ===
NAT_SC: 112
N_MODE: 336
CMAT: after get emat
=== DEBUG GET_G ===
N_MODE: 336
CMAT: after get g

And the progress is keep running for three days without any output. How can I evaluate the time I need for this job?

[mesonepigreco]

Are you sure that the v4 is required in your system? Benchmark it on a smaller cell if you can beforehand, as it is an extremely expensive calculation. We found in most cases it to be completely negligible.

With 336 modes it is very challenging to converge the v4 calculation. The scaling is quite complex. The calculation of the v4 tensors scales as N^4 * N_configs, while there is an inversion of a matrix that scales as N^6 (where N is the number of modes).

If you really need the v4, I suggest you try to compute the Hessian using the tdscha package. This has a specific implementation to avoid both these two operations (It will be much faster and also give an easy progress status to address how much time is required, however, it will still be a tough calculation).

Here there is a template input file for the tdscha package (pip install tdscha) which you can use to calculate individual frequencies, with their static hessian values and - for free- the respective dynamical poles and eventual satellites.
Note that the v4 usually requires also a lot of configuration to converge.

[jgSi1113]

Are you sure that the v4 is required in your system? Benchmark it on a smaller cell if you can beforehand, as it is an extremely expensive calculation. We found in most cases it to be completely negligible.

With 336 modes it is very challenging to converge the v4 calculation. The scaling is quite complex. The calculation of the v4 tensors scales as N^4 * N_configs, while there is an inversion of a matrix that scales as N^6 (where N is the number of modes).

If you really need the v4, I suggest you try to compute the Hessian using the tdscha package. This has a specific implementation to avoid both these two operations (It will be much faster and also give an easy progress status to address how much time is required, however, it will still be a tough calculation).

Here there is a template input file for the tdscha package (pip install tdscha) which you can use to calculate individual frequencies, with their static hessian values and - for free- the respective dynamical poles and eventual satellites. Note that the v4 usually requires also a lot of configuration to converge.

Thanks a lot.
The reason for the v4 is that I want to do similar work as https://arxiv.org/abs/2501.12068. Due to the harmonic phonon dispersions of my system show imaginary frequencies, I used the SSCHA to minimize and obtain the auxiliary SSCHA/Hessian dynamical matrices and further to check the Ionic Lattice Anharmonicity effect in the stability of the system. To be honest, after the SSCHA minimization, (without v4) the Hessian related phonons is fully stable. So can I just use this v4-freed Hessian phonon to confirm the fact of Ionic Lattice Anharmonicity induced stability?