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Supporting Information for Transferability in Machine Learning for Electronic
Structure via the Molecular Orbital Basis
Matthew Welborn, Lixue Cheng, and Thomas F. Miller III
Division of Chemistry and Chemical Engineering,
California Institute of Technology, Pasadena, CA 91125, USA
(Dated: August 6, 2018)
I. EXPANDED SMALL MOLECULE PREDICTION STATISTICS
TABLE S1. Detailed decomposition of ML predictions of CCSD correlation energies for the collection of small molecules in
Tab. II, with the number of training and testing geometries indicated. Mean and Max Errors are reported for the diagonal
(∆

d
) and off-diagonal (∆

o
) contributions to the correlation energy, as well as the corresponding total and relative correlation
energy errors.
Geometries

d
(mH)

o
(mH)
Error (mH)
Rel. Error(%)
Molecule
Train
Test
Mean
Max
Mean
Max
Mean
Max
Mean
Max
H
2
50
100
0.002
0.005
NA
NA
0.00
0.00
0.00
0.01
N
2
50
100
0.016
0.202
0.008
0.052
0.06
0.19
0.01
0.05
F
2
50
100
0.003
0.037
0.002
0.014
0.03
0.18
0.00
0.03
HF
50
100
0.002
0.050
0.005
0.064
0.03
0.23
0.01
0.08
NH
3
50
100
0.026
0.112
0.014
0.056
0.16
0.57
0.06
0.23
CH
4
50
100
0.013
0.088
0.006
0.051
0.03
0.10
0.01
0.05
CO
50
100
0.002
0.009
0.027
0.073
0.03
0.07
0.01
0.02
CO
2
50
100
0.005
0.027
0.004
0.087
0.04
0.17
0.01
0.03
HCN
50
100
0.006
0.030
0.007
0.050
0.04
0.17
0.01
0.05
HNC
50
100
0.016
0.218
0.012
0.072
0.09
0.45
0.03
0.13
C
2
H
2
50
100
0.032
0.193
0.023
0.137
0.21
0.61
0.06
0.19
C
2
H
4
50
100
0.041
0.344
0.016
0.155
0.30
0.75
0.08
0.21
C
2
H
6
50
1000
0.037
0.460
0.014
0.101
0.33
1.27
0.08
0.31
200
1000
0.028
0.413
0.011
0.108
0.21
1.22
0.05
0.30
CH
2
O
50
100
0.012
0.091
0.010
0.099
0.09
0.33
0.02
0.08
HCO
2
H
50
1000
0.036
0.288
0.020
0.341
0.40
1.24
0.06
0.19
100
1000
0.026
0.522
0.015
0.375
0.27
0.86
0.04
0.14
CH
3
OH
50
100
0.018
0.073
0.013
0.078
0.14
0.55
0.03
0.12
CH
2
F
2
50
1000
0.044
1.709
0.023
0.691
0.73
2.94
0.11
0.43
100
1000
0.035
0.724
0.018
0.560
0.56
2.05
0.08
0.30
C
6
H
6
50
100
0.023
0.235
0.009
0.170
0.30
1.19
0.03
0.12
H
2
O
cc-pVDZ
50
200
0.017
0.097
0.008
0.063
0.05
0.22
0.02
0.10
cc-pVTZ
50
200
0.014
0.077
0.010
0.050
0.04
0.14
0.02
0.05
cc-pVQZ
50
200
0.015
0.081
0.011
0.057
0.05
0.20
0.02
0.07
cc-pV5Z
50
200
0.018
0.109
0.012
0.055
0.08
0.37
0.03
0.13
Two sizes of training sets are presented to illustrate error reduction.
Results for several basis sets provided.
2
II. MACHINE LEARNING MP2 PREDICTIONS
TABLE S2. As for Tab. S1, but with MP2 correlation energies rather than CCSD.
Geometries

d
(mH)

o
(mH)
Error(mH)
Rel. Error(%)
Molecule
Train
Test
Mean
Max
Mean
Max
Mean
Max
Mean
Max
H
2
50
100
0.003
0.005
NA
NA
0.00
0.00
0.01
0.02
N
2
50
100
0.014
0.163
0.013
0.233
0.15
0.48
0.04
0.12
F
2
50
100
0.003
0.064
0.002
0.013
0.04
0.33
0.01
0.06
HF
50
100
0.003
0.047
0.004
0.059
0.03
0.28
0.01
0.10
NH
3
50
100
0.023
0.119
0.016
0.067
0.14
0.50
0.06
0.21
CH
4
50
100
0.012
0.077
0.007
0.049
0.03
0.11
0.02
0.05
CO
50
100
0.004
0.015
0.005
0.023
0.03
0.10
0.01
0.03
CO
2
50
100
0.005
0.029
0.004
0.083
0.03
0.17
0.00
0.03
HCN
50
100
0.006
0.036
0.007
0.065
0.04
0.23
0.01
0.07
HNC
50
100
0.011
0.055
0.011
0.281
0.08
0.46
0.02
0.14
C
2
H
2
50
100
0.025
0.145
0.022
0.219
0.24
0.80
0.08
0.26
C
2
H
4
50
100
0.034
0.200
0.015
0.359
0.26
0.85
0.08
0.25
C
2
H
6
50
1000
0.035
1.286
0.013
0.129
0.36
2.28
0.10
0.61
150
1000
0.022
0.416
0.010
0.102
0.17
0.97
0.05
0.26
CH
2
O
50
100
0.009
0.088
0.011
0.144
0.10
0.36
0.03
0.09
HCO
2
H
50
1000
0.031
0.281
0.020
0.357
0.36
1.67
0.06
0.26
100
1000
0.025
0.240
0.015
0.182
0.28
0.96
0.04
0.15
CH
3
OH
50
100
0.015
0.070
0.013
0.091
0.14
0.72
0.03
0.17
CH
2
F
2
50
1000
0.035
1.266
0.023
0.863
0.54
3.79
0.08
0.56
100
1000
0.024
0.461
0.018
0.512
0.35
2.18
0.05
0.32
C
6
H
6
50
100
0.021
0.151
0.009
0.197
0.34
1.20
0.04
0.13
H
2
O
cc-pVDZ
50
200
0.019
0.098
0.012
0.075
0.09
0.35
0.04
0.17
cc-pVTZ
50
200
0.018
0.084
0.014
0.069
0.10
0.42
0.04
0.16
cc-pVQZ
50
200
0.020
0.097
0.015
0.081
0.09
0.35
0.03
0.12
cc-pV5Z
50
200
0.016
0.090
0.013
0.062
0.11
0.44
0.03
0.15
Two sizes of training sets are presented to illustrate error reduction.
Results for several basis sets provided.
3
-6
-4
-2
0
2
4
6
(a)
(H
2
O)
4
,
trained
on
H
2
O
+
(H
2
O)
2
Prediction:
Mean
=
0.61
Max
=
2.1
Rel.
Mean
=
0.057%
GPR
Baseline:
Mean
=
0.77
Max
=
2.5
Rel.
Mean
=
0.072%
Prediction
Error
-6
-4
-2
0
2
4
6
(b)
(H
2
O)
5
,
trained
on
H
2
O
+
(H
2
O)
2
Prediction:
Mean
=
0.82
Max
=
2.6
Rel.
Mean
=
0.061%
GPR
Baseline:
Mean
=
0.85
Max
=
3.6
Rel.
Mean
=
0.063%
Prediction
Error
-6
-4
-2
0
2
4
6
-10
-5
0
5
10
(c)
(H
2
O)
6
,
trained
on
H
2
O
+
(H
2
O)
2
Prediction:
Mean
=
1.1
Max
=
4.0
Rel.
Mean
=
0.068%
GPR
Baseline:
Mean
=
0.98
Max
=
3.8
Rel.
Mean
=
0.061%
Prediction
Error
True
MP2
E
c
FIG. S1. As for Fig. 2, but with MP2 in place of CCSD. Parallelity error is removed via a global shift in the predicted energies
of the tetramer, pentemer, and hexamer by 0.68, 0.40, and 0.38 mH, respectively.
4
-6
-4
-2
0
2
4
6
(a) Trained on methane + ethane
Prediction (Mean, Max, Rel. Mean):
Prediction Error
Butane (1.2, 4.8, 0.16%)
Isobutane (1.3, 4.2, 0.17%)
-6
-4
-2
0
2
4
6
-4
-3
-2
-1
0
1
2
3
(b) Trained on methane + ethane + propane
Prediction (Mean, Max, Rel. Mean):
Prediction Error
True MP2
E
c
Butane (0.49, 1.7, 0.068%)
Isobutane (0.77, 2.3, 0.11%)
FIG. S2. As for Fig. 3, but with MP2 in place of CCSD. Parallelity error is removed via a global shift in the predicted energies
of butane and isobutane by (a) 32 and 21 mH and (b) 3.3 and 0.87 mH, respectively. The Mean and Max GPR baseline errors
for butane are 0.40 and 1.2 mH, respectively. For isobutane, these errors are 0.47 and 1.7 mH.
-5
-4
-3
-2
-1
0
1
2
3
4
(a)
Trained
on
CH
4
+
H
2
O
+
HCO
2
H
Prediction
(Mean,
Max,
Rel.
Mean):
Prediction
Error
HCO
2
H
(0.33,
1.3,
0.052%)
H
2
O
(0.051,
0.18,
0.020%)
CH
4
(0.067,
0.23,
0.034%)
-5
-4
-3
-2
-1
0
1
2
3
4
-5
-4
-3
-2
-1
0
1
2
3
-6
-4
-2
0
2
4
6
8
(b)
Trained
on
CH
4
+
H
2
O
+
HCO
2
H
Prediction
(Mean,
Max,
Rel.
Mean):
Prediction
Error
True
MP2
E
c
CH
3
OH
(0.57,
2.4,
0.13%)
-5
-4
-3
-2
-1
0
1
2
3
-6
-4
-2
0
2
4
6
8
FIG. S3. As for Fig. 4, but with MP2 in place of CCSD. In panel (b), parallelity error is removed via a global shift in the
predicted energy by 4.5 mH.
5
-6
-4
-2
0
2
4
-4
-3
-2
-1
0
1
2
3
Trained
on
H
2
O
Prediction
(Mean,
Max,
Rel.
Mean):
Prediction
Error
True
MP2
E
c
NH
3
(0.45,
1.5,
0.19%)
CH
4
(0.57,
1.9,
0.29%)
HF
(0.062,
0.49,
0.023%)
-6
-4
-2
0
2
4
-4
-3
-2
-1
0
1
2
3
FIG. S4. As for Fig. 5, but with MP2 in place of CCSD. Parallelity error is removed via a global shift in the predicted energies
of ammonia, methane, and hydrogen fluoride by 24, 51, and 12 mH, respectively.