This part of the guide describes the calculation of harmonic frequencies with CRYSTAL, using Cu2O as an example.

During the geometry optimization, we used the first derivatives of the energy with respect to atomic displacements to locate a stationary point on the potential energy surface (derivatives are zero). In a harmonic frequency calculation, we calculate the second derivatives of the enegy with respect to atomic displacements. This yields us the vibrational frequencies of the system and tells us the nature of the stationary point (minimum, maximum, saddle point). In CRYSTAL, the second derivatives are obtained with numerical differentiation of the analytical first derivatives. 

Additional information

Here we calculate the vibrational frequencies only at the Γ-point. In simple terms, this means that the atoms in all unit cells of the crystal are vibrating in the same phase. Later parts of the guide discuss phonon dispersion relation calculations, where the vibrational modes are sampled more extensively in the reciprocal space.  

Prerequisites

You need the file Cu2O_Pn-3m_PBE0_TZVP_opt_done.d12 created in the geometry optimization of Cu2O (section Create a CIF of the optimized structure). This input file contains the optimized geometry of Cu2O.

1) Create a frequency calculation input from the input with the optimized geometry:

cp Cu2O_Pn-3m_PBE0_TZVP_opt_done.d12 Cu2O_Pn-3m_PBE0_TZVP_fq.d12

2) Replace the FINDSYM and TESTGEOM keywords in the input with:

FREQCALC
NUMDERIV
2
ENDFREQ

NUMDERIV 2 means central differentiation, which is numerically more reliable than simple one-sided differentiation.

3) The default SCF convergence criterion in FREQCALC is 10–10 a.u. which is very tight. For a Γ-point frequency calculation the criterion can typically be relaxed a little bit. Add after TOLINTEG:

TOLDEE
9

The complete input is:

Cu2O_Pn-3m_PBE0_TZVP_fq.d12
Cu2O (Pn-3m) frequency calculation (DFT-PBE0/TZVP)
CRYSTAL
0 0 0
224
4.31764698
2
29    0.000000000000E+00   0.000000000000E+00   0.000000000000E+00
8     2.500000000000E-01   2.500000000000E-01   2.500000000000E-01
FREQCALC
NUMDERIV
2
ENDFREQ
ENDGEOM
8 7
0 0 6 2.0 1.0
  27032.382631      0.21726302465E-03
  4052.3871392      0.16838662199E-02
  922.32722710      0.87395616265E-02
  261.24070989      0.35239968808E-01
  85.354641351      0.11153519115
  31.035035245      0.25588953961
0 0 2 2.0 1.0
  12.271113873      0.39768730901
  4.9159842006      0.24627849430
0 0 1 0.0 1.0
 0.90086482370      1.0000000000
0 1 1 0.0 1.0
 0.25000000000      1.0 1.0
0 2 4 4.0 1.0
  75.300554155      0.60685103418E-02
  17.743733858      0.41912575824E-01
  5.5355828651      0.16153841088
  2.0685535103      0.35706951311
0 2 1 0.0 1.0
 0.78238772422      1.0000000000
0 3 1 0.0 1.0
 1.2000000000       1.0000000000
29 12
0 0 8 2.0 1.0
 377518.79923      0.22811766128E-03
 56589.984311      0.17688035931E-02
 12878.711706      0.91993460227E-02
 3645.3752143      0.37411016434E-01
 1187.0072945      0.12189873737
 426.46421902      0.28983900714
 165.70660164      0.41531872174
 65.598942707      0.21905799287
0 0 4 2.0 1.0
  414.41265811     -0.24682525053E-01
  128.32056039     -0.11716827406
  20.622089750      0.55301315941
  8.7821226045      0.52242718609
0 0 2 2.0 1.0
 13.741372006      -0.22736061821
 2.2431246833      0.71761210873
0 0 1 1.0 1.0
 0.89370549079     1.0000000000
0 0 1 0.0 1.0
  0.35              1.0000000000
0 1 1 0.0 1.0
  0.14              1.0 1.0
0 2 6 6.0 1.0
 2034.7596692       0.23524822298E-02
 481.90468106      0.19134070751E-01
 154.67482963      0.90171105278E-01
 57.740576969      0.26063284735
 23.099052811      0.42093485770
 9.3882478591      0.24344615121
0 2 3 6.0 1.0
 37.596171210      -0.28991094530E-01
 5.1240690810      0.54919083831
 2.0119996085      0.93793330488
0 2 1 0.0 1.0
 0.73860686002     1.0000000000
0 3 4 10.0 1.0
 74.129460637      0.14363216676E-01
 21.359842587      0.86628177096E-01
 7.4995240537      0.25631430541
 2.7601394169      0.40374062368
0 3 1 0.0 1.0
 0.95362061236     0.39427042447
0 3 1 0.0 1.0
 0.28420862520     0.23091146816
99 0
ENDBAS
DFT
PBE0
ENDDFT
SHRINK
0 8
8 8 8
TOLINTEG
8 8 8 8 16
TOLDEE
9
MAXCYCLE
100
FMIXING
80
EXCHSIZE
30000000
BIPOSIZE
30000000
END

Running the frequency calculation

Submit the job normally using jsub:

jsub -np 8 crystal Cu2O_Pn-3m_PBE0_TZVP_fq.d12

Analyzing the results

An example output file is available here: Cu2O_Pn-3m_PBE0_TZVP_fq_only.o.

The vibrational modes can be visualized for example with Jmol or CRYSPLOT (see Aalto Solid State Chemistry Wiki).

The output file lists the vibrational frequencies as follows: :

    MODES         EIGV          FREQUENCIES     IRREP  IR   INTENS    RAMAN
             (HARTREE**2)   (CM**-1)     (THZ)             (KM/MOL)
    1-   3   -0.7906E-21      0.0000    0.0000  (F1u)   A (     0.00)   I
    4-   6    0.1617E-06     88.2545    2.6458  (F2u)   I (     0.00)   I
    7-   8    0.2594E-06    111.7855    3.3512  (Eu )   I (     0.00)   I
    9-  11    0.4435E-06    146.1578    4.3817  (F1u)   A (     0.00)   I
   12-  12    0.2447E-05    343.3387   10.2930  (Au )   I (     0.00)   I
   13-  15    0.5124E-05    496.7951   14.8935  (F2g)   I (     0.00)   A
   16-  18    0.7994E-05    620.5167   18.6026  (F1u)   A (     0.00)   I

Modes 1–3 with zero frequency at the Γ-point are translational (acoustic) phonon modes. Modes 4–18 with non-zero frequencies are optical modes. For 3D solids, we always have 3 acoustic modes and 3N−3 optical phonon modes. There are many degenerate vibrational modes with the same frequency due to the high symmetry of the Cu2O crystal.

Optical modes 9–11 and 16–18 are IR-active. Modes 13–15 are Raman active. We will calculate the intensities for these modes later.

A thermodynamic analysis is printed under THERMODYNAMIC FUNCTIONS WITH VIBRATIONAL CONTRIBUTIONS, but normally Γ-point frequencies are not enough for accurate thermodynamics (full phonon dispersion relations are needed). Total Gibbs free energy is the term EL+E0+ET+PV-TS.


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