True Geometry Minimum

The problem

After optimizing the geometry of a molecular system or lanthanide complex and verifying that the norm of the gradient has been sufficiently reduced, users tend to be satisfied with the geometry obtained and proceed to compute other properties.

Is this warranted?


Actually it is very important to determine if this geometry corresponds to a true minimum in the potential energy hypersurface where the atoms move.


The solution

By carrying out a FORCE calculation after having minimized the geometry and verifying that there are no negative force constants.

And what if there are negative force constants?

Then you must eliminate them, by carrying out IRC calculations and displacing the geometry in either direction of the corresponding normal coordinate.

Subsequently, at the displaced geometries, you reoptimize them; carry out a FORCE calculation again; and repeat this cycle until you no longer have any negative force constants.

This process is thoroughly explained in the tutorial below.


When one optimizes the geometry of a molecular system, such as a lanthanide complex, MOPAC will attempt to reduce the gradient norm. For lanthanide complexes, we normally recommend stopping the geometry optimization when the gradient norm becomes smaller than 0.25 kcal/Ångstrom, via the keyword GNORM=0.25. 

However, the simple zeroing of the gradient norm does not guarantee that the corresponding geometry is stable; that it is at a true energy minimum.

To clarify this point, let us consider the equivalent problem in one dimension, when the gradient is the function derivative.

As is well known, when the derivative of a one-variable function is zero, that does not automatically mean that the function is at a minimum. It simply means that the derivative is zero, and that could be one of three situations: the function could be at a minimum, at a maximum, or at an inflection point, as shown below.

function minimum point sparkle qtiplotfunction maximum point sparkle qtiplotfunction inflection point sparkle qtiplot

Thus, in order to decide among the three situations, one must calculate the second derivative, which is positive if the function is at a minimum, negative at a maximum, or zero, at either a flat surface or an inflection point.

In a molecular system, such as a lanthanide complex, with N atoms, there are 3 Cartesian coordinates per atom, for a total of 3N Cartesian coordinates.

The movements of the atoms, within the linear approximation, can be decoupled via diagonalization of the Hessian matrix, by a unitary transformation of the Cartesian coordinates into a new set of coordinates called normal coordinates. The normal coordinates are linear combinations of the Cartesian coordinates of the atoms, and correspond to molecular vibrations, rotations and translations.

Of the 3N normal coordinates, 3 correspond to translations of the whole molecular system or lanthanide complex along the directions x, y, and z.

For a linear molecule, two other normal coordinates correspond to rotations of the whole molecular system or lanthanide complex along the main axis of symmetry, whereas for nonlinear molecules there are three other normal coordinates corresponding to rotations.

The remaining normal coordinates correspond to vibrations.

As such, for linear molecules there are 3N-2 vibrations, and for nonlinear molecules, there are 3N-3 vibrations.

The second derivatives of the energy with respect to each normal coordinate give the force constants. Thus, if the gradient norm is zero and the force constant is positive (for a given normal coordinate), the geometry is at a true minimum with respect to that normal coordinate.

On the other hand, if the gradient norm is zero and the force constant is negative for a given normal coordinate, the geometry is at a maximum with respect to that normal coordinate.

Finally, if the gradient norm is zero and the force constant is also zero, either the normal coordinate is at an inflection point, or it is lying on a flat surface. And such is the case for translations and rotations. 

sparkle flat surface

Indeed, since there are no impediments to translations or rotations, the force constants correponding to these movements should be zero.

As an example, below we represent the three normal coordinates for the translations of the flat molecule of LuBr3, where the blue arrows indicate the directions of the movements of each atom along the corresponding normal coordinate.

lutetium translations


And below, we also represent the normal coordinates corresponding to the three rotations of the same molecule.

lutetium rotations


On the other hand, there are internal forces in every molecular system or lanthanide complex, and, therefore, the force constants corresponding to vibrations are not zero.

Below, we present three of the six normal coordinates of LuBrcorresponding to vibrations.

lutetium vibrations


The vibrational frequencies, measured by infrared or Raman spectroscopy are proportional to the square root of the force constants.

So, when one of the vibrational force constants is negative, the corresponding vibrational frequency is imaginary.

A valid optimized geometry must have all vibrational force constants positive, in which case one can safely say that the geometry is sitting at a true minimum at the potential energy hypersurface.



Is the optimized geometry a true minimum?

As a case study, let us consider complex OGIJAP [aqua-bis(nitrilotriacetato-N,O,O',O'')-holmium(iii)] below:

ogijap holmium sparkle


In GABEDIT, we drew its structure, and then run MOPAC to optimize the geometry. We then optimized the geometry.

The output looks normal, as shown below.

 CYCLE:   218 TIME:   0.053 TIME LEFT:  2.00D  GRAD.:     0.255 HEAT: -684.9659    
 CYCLE:   219 TIME:   0.053 TIME LEFT:  2.00D  GRAD.:     0.256 HEAT: -684.9659    
 CYCLE:   220 TIME:   0.052 TIME LEFT:  2.00D  GRAD.:     0.270 HEAT: -684.9659    
 CYCLE:   221 TIME:   0.054 TIME LEFT:  2.00D  GRAD.:     0.282 HEAT: -684.9660    
 CYCLE:   222 TIME:   0.056 TIME LEFT:  2.00D  GRAD.:     0.321 HEAT: -684.9661    


 Mopac file generated by Gabedit


As such, in principle one would not suspect that there is anything wrong with the geometry, nor by the output above, neither by looking at a picture of the converged geometry (see below).

holmium complexes sparkle


At this point, we must verify whether or not this geometry is a true minimum. Therefore, we need to run a FORCE calculation in order to check whether or not all force constants are positive. For convenience, we provide the .mop input file ogijap_opt_force.mop

As a result of the MOPAC calculation, two output files are produced: ogijap_opt_force.out and ogijap_opt_force.aux. The .aux file contains all force constants, including translations and rotations. On the other hand, the .out file shows only the vibrations.

Here is a sample of the .aux file for this calculation:

 #                                  #
 #      Normal mode analysis        #
 #                                  #
 ROTAT_CONSTS:CM(-1)[3]=     0.007339     0.004886     0.004735
 PRI_MOM_OF_I:10**(-40)*GRAM-CM**2[3]  3814.203258  5729.543526  5911.848790
  -139.72    36.84    46.29    56.84    59.40    64.38    79.30    82.57   117.29   119.35
   127.31   137.39   155.41   168.66   179.52   187.85   194.90   202.94   213.81   250.49
   273.24   285.42   288.57   299.52   307.42   313.25   326.83   335.42   344.27   351.61
   380.88   389.69   394.38   403.56   421.56   443.92   481.39   487.52   531.96   537.37
   541.75   548.57   556.87   580.86   588.39   623.44   626.16   637.05   639.00   694.95
   697.80   709.55   714.79   727.07   730.10   773.33   885.18   939.80   943.25   950.85
   956.68   963.23   969.19   990.60   994.31  1024.13  1026.79  1027.89  1031.55  1078.07
  1081.06  1102.59  1104.51  1127.39  1128.34  1152.63  1153.53  1221.08  1221.33  1236.90
  1238.89  1251.49  1253.09  1265.49  1267.18  1272.07  1275.72  1295.58  1296.52  1325.93
  1330.48  1337.18  1339.65  1361.21  1373.99  1376.52  1384.17  1415.56  1422.62  1429.82
  1773.40  1779.17  1792.04  1796.08  1803.07  1839.70  2423.60  2550.13  2649.68  2650.10
  2658.51  2661.38  2665.12  2667.64  2727.41  2729.27  2732.51  2733.83  2734.50  2736.18
     0.10     0.11     0.11    12.34     8.45     8.72

Please, note that the last six frequencies are close to zero, and they represent the rotations and translations. All the other frequencies correspond to vibrations, and are all positive, except the first one. Indeed, there appeared a negative frequency: -139.72. Actually it is not really negative - its is an imaginary frequency due to a negative force constant. The output shows it as negative instead of indicating it as imaginary, because it is much simpler to print the result that way.

So, we have a problem: an imaginary frequency appeared!

But before we go solve this problem, let us examine the corresponding output: the .out file.

          HEAT OF FORMATION =    -682.326855 KCALS/MOLE

          ZERO POINT ENERGY     172.114 KCAL/MOL


           NORMAL COORDINATE ANALYSIS (Total motion = 1 Angstrom)

   Root No.       1         2         3         4         5         6         7         8

                 1 A       2 A       3 A       4 A       5 A       6 A       7 A       8 A   

                -139.7      36.8      46.3      56.8      59.4      64.4      79.3      82.6
           1    0.0039   -0.0305    0.0116   -0.0027    0.0200   -0.0099   -0.0022   -0.0013
           2   -0.0038    0.0183   -0.0042   -0.0045   -0.0172   -0.0087    0.0052   -0.0139
           3    0.0009   -0.0230   -0.0170    0.0016   -0.0388    0.0694   -0.0286    0.0577
           4    0.0041   -0.0303    0.0105   -0.0030    0.0174   -0.0065   -0.0022    0.0013

The .out file has a different format. It shows columns containing the frequencies on top and, underneath, the coefficientes of the normal coordinates for each of the atoms cartesian x, y, z coordinates. Note that the output recognizes the presence of the imaginary frequency and identifies the system as not being a ground state.

One could not have known that without running the FORCE calculation.

So this geometry is not a true minimum.

Since this geometry has one imaginary frequency, it is actually sitting on top of a maximum in the energy hypersurface.

In order to remove it, we must distort the geometry in either direcion of the corresponding normal coordinate.


irc imaginary frequencies

In order to run an IRC calculation and tell MOPAC to distort the geometry to either side of the maximum, we must use the keyword IRC=1* .

Even if there are more than one imaginary frequency, one should always distort the geometry in both directions of the normal coordinate of the first imaginary frequency, which is always the most negative of all. This way, there is a good chance that by eliminating the first, one may also end up eliminating other imaginary frequencies at the same time.

For convenience, we provide the corresponding input .mop file ogijap_IRC.mop.

After running the calculation, a .xyz file is created. It contains a series of geometries distorted to either side of the maximum. For our purposes, we pick the first and the last ones, and ignore the others. These will correspond to geometries A' and B' in the figure above.

For convenience, we provide files and ogijap_b_prime.mop.

Then, we reoptimize both A' and B' geometries. We examine the corresponding .out files and pick the one with the smallest value for the heat of formation.

In this case, the A' geometry led to the lowest energy A geometry ogijap_a_prime.arc.

By running a FORCE calculation again on the A geometry, we verify that there are no more imaginary frequencies.

Here is a sample of the .aux file for this calculation:

 #                                  #
 #      Normal mode analysis        #
 #                                  #
 ROTAT_CONSTS:CM(-1)[3]=     0.007260     0.004894     0.004779
 PRI_MOM_OF_I:10**(-40)*GRAM-CM**2[3]  3856.008931  5719.576917  5857.754348
    32.25    43.64    45.49    53.83    64.02    78.91    82.84   111.56   114.79   123.61
   130.95   151.59   158.35   169.72   184.01   190.83   195.00   207.94   225.48   246.04
   269.12   287.39   297.56   305.35   309.60   316.55   333.53   342.68   348.81   360.25
   384.31   389.21   398.23   404.29   423.67   448.90   494.01   497.95   516.66   527.39
   533.69   548.11   550.61   573.06   575.06   624.50   629.82   632.69   638.26   694.80
   697.56   707.99   713.54   726.23   727.73   783.30   867.08   938.77   943.64   951.71
   955.83   962.96   966.91   995.14  1004.32  1027.43  1029.93  1035.49  1039.64  1084.06
  1084.50  1101.81  1104.51  1135.05  1136.60  1146.61  1149.52  1223.06  1226.33  1239.87
  1242.14  1249.03  1250.47  1267.86  1269.31  1276.14  1276.79  1292.89  1295.65  1325.63
  1329.92  1341.57  1342.79  1374.73  1377.76  1382.03  1387.38  1395.79  1425.76  1429.94
  1771.76  1775.03  1789.46  1793.21  1800.34  1837.18  2274.16  2402.77  2637.86  2650.42
  2657.61  2659.51  2660.51  2663.07  2726.70  2728.11  2730.73  2731.54  2731.91  2734.50
     0.10     0.11     0.12    12.31     8.42     8.32

So, the A geometry of this complex, shown below, is thus verified to be at a true minimum.

ogijap sparkle optimiz