Supplementary information for
“The chromium dimer: closing a chapter of quantum chemistry”
Henrik R. Larsson,
1, 2
Huanchen Zhai,
1
C. J. Umrigar,
3
and Garnet Kin-Lic Chan
1,
a)
1)
Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, CA 91125,
USA
2)
Department of Chemistry and Biochemistry, University of California, Merced, CA 95343,
USA
b)
3)
Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY 14853,
USA
S1. ELECTRONIC STRUCTURE
Our simulations for the electronic structure of the
X
1
Σ
+
g
Cr
2
ground state were based on the com-
posite method presented in the main text. It con-
sists of an estimate of the exact binding energy
at the cc-pVDZ-DK basis set level
1
(correlating 28
electrons in 76 orbitals), corrected for the basis-set
limit (CBS) by the second order matrix-product-state-
approximated multireference retaining the excitation
degree perturbation,
2–5
MPS-REPT2, level of theory.
All computations were based on the spin-free exact-
two-component Hamiltonian,
6,7
and used
PySCF
,
8,9
block2
,
5,10
and
Arrow
.
11–13
The composite binding
energies,
∆
E
(
R
) =
E
(
R
)
−
E
(
∞
)
, at each interatomic
distance
R
, were calculated by
∆
E
=∆
E
PDZ
(
“exact”
) + ∆
E
CBS
(
MPS-REPT2
)
−
∆
E
DZ
(
MPS-REPT2
)
,
(S1)
where we have omitted the
R
-dependence, for sim-
plicity. To compare with experiment, following Li et
al.,
12
we subtracted a zero-point energy of
0
.
029 eV
from the experimental PEC.
A. PDZ energies
The estimates of the exact energies at the cc-pVDZ-
DK basis set level were computed as the average of en-
ergies from two distinct methods. The first method is
selected heat bath configuration interaction plus per-
turbation theory (SHCI), which we took directly from
Ref. 12. Using extrapolation this provides an estimate
of the exact energy in the basis; the used data is shown
in Table S1. For calculating the binding energies, the
energy of the Cr atom was taken from the SHCI cal-
culations with a value of
−
1049
.
93257 E
H
, which is in
very good agreement with a previous DMRG estimate
of
−
1049
.
93254 E
H
.
14
For the most difficult case, the
optimization at
R
= 2
.
25
Å, the variational part of the
SHCI computation took about 60 hours on 2 nodes
each with 80 Intel Xeon E7-8880 CPUs. Computing
a)
Electronic mail: garnetc [a t] caltech . edu
b)
Electronic mail: larsson22_cr2 [a t] larsson-research. de
TABLE S1. SHCI energies in
E
H
at the cc-pVDZ-DK
level. Shown are the best variational, perturbative, and
extrapolated energies in
E
H
. The energies are shifted by
2099 E
H
.
R/
Å
E
(
var
)
E
(
pt
)
E
(
extrap.
)
1
.
50
−
0
.
8761
−
0
.
8901
−
0
.
8928
1
.
55
−
0
.
8907
−
0
.
9053
−
0
.
9084
1
.
60
−
0
.
8984
−
0
.
9137
−
0
.
9173
1
.
68
−
0
.
9019
−
0
.
9185
−
0
.
9229
1
.
80
−
0
.
8981
−
0
.
9165
−
0
.
9221
2
.
00
−
0
.
8889
−
0
.
9101
−
0
.
9178
2
.
25
−
0
.
8827
−
0
.
9046
−
0
.
9126
2
.
50
−
0
.
8813
−
0
.
9020
−
0
.
9096
2
.
75
−
0
.
8775
−
0
.
8973
−
0
.
9049
3
.
00
−
0
.
8722
−
0
.
8909
−
0
.
8984
3
.
25
−
0
.
8658
−
0
.
8840
−
0
.
8908
the perturbative correction took 15 hours using one
node of the same configuration.
The second method is spin-adapted DMRG, com-
puted using
block2
.
10
We used natural orbitals ob-
tained by diagonalizing the spin-summed unrestricted
coupled cluster singles and doubles (UCCSD) density
matrix. We optimized the orbital ordering with a ge-
netic algorithm
15
at
R
= 2
Å. The accuracy of the
DMRG is controlled by a single parameter, the bond
dimension
D
. The DMRG optimizations used a max-
imum bond dimension between
D
max
= 18
,
000
and
28
,
000
, depending on the convergence at each bond
distance. One DMRG iteration (sweep) at
D
max
=
28
,
000
took around 20 hours on five nodes with 24-
core Intel Xeon 8276 CPUs. For the most difficult
case, the optimization at
R
= 2
.
25
Å, nine iterations
were required for convergence.
To extrapolate to infinite bond dimension, we fol-
lowed the standard procedure
15
and did a linear ex-
trapolation of the energy as function of discarded
weight. For the linear fit we use data points ob-
tained from a backward propagation, starting with
the largest bond dimension and successively reopti-
mizing the matrix product state at smaller bond di-
mensions. The energy is a linear function of discarded
weight in the limit of small discarded weight. However
at the largest bond dimension, the wavefunctions are
imperfectly optimized (as they do not have informa-
tion from states with much larger bond dimension),
while at the smallest bond dimension, the discarded
weight may be too large to be in the linear regime.
2
Consequently, we discard the data of the largest bond
dimensions and smallest bond dimensions if it signifi-
cantly worsens
R
2
. To account for outliers, we used a
robust least-squares fit with a Huber loss function.
16
The raw and extrapolated DMRG energies, together
with the bond dimensions used for the optimizations
and for the fits are shown in Table S2.
It should be emphasized that the SHCI and DMRG
extrapolations are ultimately uncontrolled proce-
dures; the method of extrapolation is not unique and
estimating the error of the extrapolation is difficult,
and this is the importance of having two different esti-
mates. However, the estimated extrapolated energies
are in good agreement with each other. Note also that
the assignments of vibrational quantum numbers do
not change for reasonable variants of the extrapolation
procedure.
B. Basis set correction
To account for the basis-set correction, we use the
second order uncontracted multireference retaining
the excitation degree perturbation theory (MRREPT)
wavefunction, approximated by a matrix product
state (MPS). (Although this is strictly a matrix prod-
uct state perturbation theory,
3
this approximation
was termed matrix product state linearized coupled
cluster theory (MPS-LCC) in the literature, because
of a connection to coupled cluster theory in the limit of
a single reference theory.
4
However, to avoid confusion
with other methods also referred to as multireference
linearized coupled cluster,
17,18
we refer to the method
as MPS-REPT2 in the following). We computed the
MPS-REPT2 energies using
block2
.
5,10
We used the
cc-pV
N
Z-DK basis set, which includes up to
i
-type
functions for
N
= 5
.
19
While the basis set superposi-
tion error (BSSE) can be very large in Cr
2
for small
bases,
20,21
we did not employ any approximate BSSE
correction because we include up to quintuple zeta
bases and extrapolate to the basis set limit, where the
BSSE is zero, by definition. BSSE corrections have
been omitted in other studies as well.
12,22,23
The multireference simulations were based on a
complete active space self-consistent field (CASSCF)
reference with a standard valence CAS(12,12) con-
sisting of 12 electrons and 12 orbitals (3d and 4s),
which can be described by 28784 spin-adapted config-
uration state functions. We optimized the CASSCF
wavefunction using the PySCF program package.
8,9
The MPS-REPT2 computations were based on the
aforementioned CAS(12,12) and, in addition, corre-
lated the eight 3s and 3p orbitals (internal space) with
up to two-fold excitations into the active and external
spaces. The MPS used to approximate the MRREPT2
wavefunction included
U
(1)
symmetry-adaptation for
N
el
and
S
z
and two large sites at either end to describe
the internal and external space.
5
We used a bond di-
mension of up to
D
= 16
,
000
, without further extrap-
olation of the energy. The PECs were generated from
the binding energies as obtained by subtracting twice
the atomic energy (the MRREPT theory is fully size
consistent). For the largest basis, one MPS-REPT2
iteration (sweep) at
D
= 16
,
000
took around 7 hours
on one node with 56 Intel Xeon 8276 CPUs. Start-
ing from a good guess, for example from a previously
computed bond distance, fewer than six iterations are
typically required to reach convergence.
a. MPS-REPT2 PEC
For comparison, the raw
MPS-REPT2 PECs for various basis sets are shown
in Figure S1. The CBS limit is obtained as shown in
the next Section, S1 C. Note that the convergence with
the basis set size is not monotonic and the cc-pVDZ-
DK binding energies are lower than the cc-pVTZ-DK
energies. This non-monotonic basis set convergence
has often been observed for Cr
2
.
5,12
Due to error can-
cellation, the cc-pVDZ-DK PEC qualitatively agrees
with the experimental curve but predicts a too large
equilibrium distance. The TZ PEC exhibits a notica-
ble double minimum whereas the CBS PEC does not.
This is related to the unusually difficult basis set con-
vergence, and has also been observed elsewhere, e.g.,
in Refs. 20,23. In contrast to other multi-reference
perturbation theories based on a CAS(12,12) refer-
ence, the raw MPS-REPT2 CBS curve shows surpris-
ingly good agreement with the experimental Casey-
Leopold curve as well as the rederived experimental
curve from this work, with a maximal error
0
.
13 eV
,
indicating its usefulness as basis set and dynamical
correlation correction.
C. Basis set extrapolation
We calculated
∆
E
CBS
(
MPS-REPT2
)
by fitting the
binding energies to
∆
E
(
N
) = ∆
E
CBS
+
A
(
N
+ 1
/
2)
−
3
,
(S2)
where
N
is the cardinal number of the basis set. We
use triple, quadruple, and quintuple zeta bases for the
fit.
D. Error estimates
Equation S1 contains error contributions from three
main sources: (1) The PDZ energy for the cc-pVDZ-
DK basis set,
∆
E
PDZ
, (2) the basis set extrapolation
energy,
∆
E
CBS
(
MPS-REPT2
)
, and (3) the error from
the approximate nature of the MPS-REPT2 method.
Additional errors that we neglect here come from,
e.g., the approximate nature of the relativistic two-
component Hamiltonian (which is, however, mostly
addressed by the basis set extrapolation because it
is exact in the complete basis set limit), non-Born-
Oppenheimer corrections, and the spline interpolation
error (because the MPS-REPT2 energies were com-
puted at different points but on a denser grid than the
SHCI energies). We calculate the total error assuming
that the individual contributions are independent (see
below).
3
TABLE S2. DMRG energies at the cc-pVDZ-DK level.
D
max
is the maximal bond dimension used for the optimization.
D
fit
min
(
D
fit
max
) are the minimal (maximal) bond dimensions used for the fit to extrapolate the energy to infinite bond
dimension,
E
(
∞
)
. The energies are shifted by
2099 E
H
.
R
2
is the coefficient of determination of the linear fit.
R D
max
D
fit
min
D
fit
max
E
(
D
max
)
E
(
∞
)
E
(
D
max
)
-
E
(
∞
)
R
2
/
Å
/
E
H
/
E
H
/
mE
H
1
.
50 18 000 6000 17 000
−
0
.
8852
−
0
.
8929
8
0
.
998
1
.
55 18 000 8000 18 000
−
0
.
9001
−
0
.
9084
8
0
.
998
1
.
60 18 000 8000 18 000
−
0
.
9083
−
0
.
9177
9
0
.
998
1
.
68 25 000 8000 22 000
−
0
.
9150
−
0
.
9227
8
0
.
997
1
.
80 25 000 10 000 24 000
−
0
.
9134
−
0
.
9221
9
0
.
995
2
.
00 25 000 10 000 22 000
−
0
.
9074
−
0
.
9183
11
0
.
992
2
.
25 28 000 14 000 27 000
−
0
.
9063
−
0
.
9131
7
0
.
998
2
.
50 18 000 8000 17 000
−
0
.
9014
−
0
.
9096
8
0
.
997
2
.
75 18 000 6000 17 000
−
0
.
8969
−
0
.
9047
8
0
.
995
3
.
00 18 000 6000 16 000
−
0
.
8906
−
0
.
8979
8
0
.
995
3
.
25 18 000 6000 16 000
−
0
.
8838
−
0
.
8911
8
0
.
991
−
1.6
−
1.4
−
1.2
−
1
−
0.8
−
0.6
−
0.4
−
0.2
0
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
3.2
binding energy/eV
bond length/Å
Casey/Leopold: Expt. fit
This work: Expt. new fit
MPS-REPT2 DZ
TZ
QZ
5Z
CBS
FIG. S1. Potential energy curves of the chromium dimer. The blue curve show the experimental result from Casey and
Leopold.
24
The dotted purple curve is our new fit based on the reassigned experimental data. The other curves show
MPS-REPT2 results in different bases and for the basis set limit (CBS).
We now discuss how we calculate the individual er-
ror estimates. (1) We estimated the error bar of the
PDZ energy to be half the absolute difference of the
DMRG and the SHCI energies, which is identical to
the standard deviation of the mean of two separate
measurements.
25
Note that both SHCI and DMRG are
methods that can systematically approximate and ex-
trapolate to the exact energy but work in a very differ-
ent way, thus the estimates of the energy are roughly
independent. (2) We compute the basis set extrapola-
tion error from the standard deviation
σ
CBS
obtained
by the least-squares fit of Eq. S2. A more conserva-
tive alternative is to define the error bar as the differ-
ence between the extrapolation with and without the
largest basis set. This gives an error that is slightly
(
∼
0
.
002 eV
) larger but which follows the same trend
as that computed from the fit. (3) We conservatively
estimated the error bar of the MPS-REPT2-based cor-
rection by taking half the absolute difference between
the values of the UCCSD(T) correction from Li et
al.
12
and that from MPS-REPT2. Due to the poor
quality of the UCCSD(T) PECs we believe that this
overestimates the error bar. An alternative would be
to estimate the error as some fraction, e.g.
∼
20 %
of
the correction. For
R
→ ∞
, the binding energy ap-
proaches zero, which also holds for this definition of
the error. This type of error estimate leads to errors
that are a factor of
∼
2
smaller than the UCCSD(T)-
estimated errors. Another alternative would be to
compare to the MPS-REPT3-based third order per-
turbative correction, which we computed as well. For
R
≥
2
Å, the error based on
20 %
of the correction and
the MPS-REPT3-based error are roughly similar and
for
R <
2
Å the MPS-REPT3-based error gets too
large, larger than even the UCCSD(T)-based error.
The problematic behavior of third order perturbation
theory at
R <
2
Å has also been seen in n-Electron
Valence State Perturbation-Theory (NEVPT3).
26
Assuming that the individual errors are indepen-
4
dent, the total error is then given as
25
ε
=
q
ε
2
PDZ
+
ε
2
CBS
+
ε
2
REPT
,
(S3)
with the individual error contributions as defined
above,
ε
PDZ
=
|
∆
E
(
SHCI
)
−
∆
E
(
DMRG
)
|
/
2
,
(S4)
ε
CBS
=
σ
CBS
,
(S5)
ε
REPT
=
|
∆
E
(
MPS-REPT2
)
−
∆
E
[
UCCSD(T)
]
|
/
2
.
(S6)
S2. VIBRATIONAL SPECTRUM
We simulated the vibrational spectrum of
52
Cr
52
Cr
and
50
Cr
52
Cr by solving the
J
= 0
vibrational
Schrödinger equation using the sine discrete vari-
able representation (DVR)
27
with 1000 grid points
placed between
∼
1
.
40
Å and
∼
3
.
45
Å. We repre-
sented the potential by cubic spline interpolation. As
atomic masses we used
51
.
9405115 u
for
52
Cr
and
49
.
9460495 u
for
50
Cr
.
28
S3. POTENTIAL ENERGY CURVE FIT
To obtain an analytical PEC based on the available
vibrational data, we solved the inverse Schrödinger
equation using 180 DVR grid points and we expressed
the PEC using the expanded Morse oscillator (EMO)
form:
29–33
V
EMO
(
R
) =
D
{
1
−
exp[
−
β
(
R
)(
R
−
R
e
)]
}
2
,
(S7)
β
(
R
) =
N
p
X
i
=0
a
i
[(
R
q
−
r
q
r
)
/
(
R
q
+
r
q
r
)]
i
.
(S8)
For the fit we used a polynomial order of
N
p
= 4
for
the evaluation of
β
(
R
)
and chose
q
= 5
. We kept
R
e
fixed at the experimental value of
1
.
6788
Å. All other
values, including
r
r
, were optimized.
Following Casey and Leopold, we included in the fit
the first nine vibrational levels and the 20 vibrational
levels starting at
4880 cm
−
1
.
24
Likewise, we did not
include the two levels at
4290 cm
−
1
and
4570 cm
−
1
in
the fit, as those are difficult to assign and Casey and
Leopold suggested that the transitions may not stem
from the
1
Σ
+
g
Cr
2
ground state. We solved the in-
verse Schrödinger equation using least-squares fitting
with the Jacobian obtained by automated differenti-
ation using the JAX library.
34
We note that many
different potential energy curves (different fit results)
lead to an excellent agreement with the experimen-
tal levels. There is hence a large uncertainty in the
derived experimental PEC. Here we chose a fit that
both has an excellent agreement with the experimen-
tal levels and that is sufficiently smooth. Including
the two
4290
/
4570 cm
−
1
levels in the fit resulted in a
PEC with small wiggles in the region of
2
.
2
–
3
.
0
Å but
did not lead to changes in the overall shape. The final
values of the EMO potential are listed in Table S3.
TABLE S3. Parameter values of the expanded Morse os-
cillator fit, Equation S8. The cluster of vibrational levels
at
4880 cm
−
1
are assigned to
v
= 23
−
42
.
Parameter
value
D/
cm
−
1
12 959
.
4
r
r
/
Å
2
.
794 295
a
0
/
Å
−
1
0
.
836 801
a
1
/
Å
−
2
−
0
.
270 609
a
2
/
Å
−
3
0
.
787 853
a
3
/
Å
−
4
−
0
.
110 491
a
4
/
Å
−
5
1
.
684 039
S4. VIBRATIONAL ASSIGNMENT AND PEC
EVALUATION
The experimental vibrational levels are compared
with our new EMO fit and our simulated PEC, which
is based on our composite method, in Table S4. Casey
and Leopold’s original assignment
24
of the cluster
starting at
4880 cm
−
1
was obtained by comparing the
experimental vibrational energies to those obtained
from a slightly modified Rydberg-Klein-Rees poten-
tial for various assignments for the vibrational lev-
els. Casey and Leopold stated that
v
= 22
–
41
was
the lowest vibrational numbering that leads to agree-
ment between the observed and the vibrational lev-
els obtained via a PEC generated from a modified
Rydberg-Klein-Rees (RKR) method.
35–38
The assign-
ment of
v
= 24
–
43
gave a good agreement also with
the two levels at
4290 cm
−
1
and
4570 cm
−
1
, that
were assigned to
v
= 18
and
v
= 21
, respectively,
based on the obtained RKR potential. However, they
stated that these transitions may not stem from the
1
Σ
+
g
Cr
2
ground state. They did a further check by
comparing the isotope shift of seven levels between
5240 cm
−
1
and
6000 cm
−
1
. The observed isotope shift
was
24
±
8 cm
−
1
whereas that of the modified RKR
potential is
36 cm
−
1
. Only when assigning the higher
levels as
v
= 20
–
39
could they obtain a potential
with an isotope shift within experimental uncertainty
(
31 cm
−
1
), but the resulting potential had a large er-
ror for the
v
= 9
level. Similar to that of the modified
RKR potential, our simulated composite curve also
results in an average isotope shift of
36 cm
−
1
in this
region. The newly fitted EMO potential results in a
slightly lower average isotope shift of
35 cm
−
1
.
5
TABLE S4: Measured and simulated vibrational energy levels. Units are in
cm
−
1
. The experimental data for
v
= 1
is from Ref. 39. All other exp. data is from Ref. 24. ZPE stands for zero point energy, relative to a potential with
minimum energy of
0
. The values for
v >
0
are relative to the ground state energy.
̃
ν
o
(
̃
ν
n
) are the vibrational levels
for the old (new) assignment for the states starting at
4880 cm
−
1
and for the state at
4290 cm
−
1
. The assignment of
the exp. levels in parentheses is not fully clear and these levels may not belong to the Cr
2
ground state.
24
̃
ν
EMO
are the
levels for the extended Morse oscillator fit based on the new assignment, and
̃
ν
theory
are the levels from our composite
theory. The difference between theory and experiment is an order of magnitude smaller for the new assignment than for
the old assignment and is comparable to the experimental uncertainty.
v
̃
ν
o
exp
̃
ν
n
exp
̃
ν
EMO
̃
ν
n
exp
−
̃
ν
EMO
̃
ν
theory
̃
ν
o
exp
−
̃
ν
theory
̃
ν
n
exp
−
̃
ν
theory
ZPE
236.15
242.77
1 452.34
±
0.02
same
452.34
0
467.84
-16
2 875
±
10
same
877.6
-3
900.7
-26
3 1280
±
10
same
1275.2
5
1293.6
-14
4 1645
±
10
same
1644.7
0
1652.1
-7
5 1985
±
15
same
1985.4
-0
1980.1
5
6 2300
±
15
same
2296.5
3
2278.7
21
7 2580
±
20
same
2577.1
3
2548.0
32
8 2830
±
20
same
2826.1
4
2790.1
40
9 3040
±
20
same
3042.9
-3
3005.0
35
10
3228.2
3193.7
11
3386.3
3358.1
12
3525.3
3502.2
13
3654.1
3631.6
14
3778.2
3752.9
15
3900.3
3871.0
16
4021.4
3988.5
17
4142.2
4106.5
18 (4290
±
20)
4262.9
(27)
4225.6
(64)
19
(4290
±
20) 4383.8
(-94)
4346.2
(-56)
20
4504.9
4468.5
21 (4570
±
20)
(same)
4626.3
(-56)
4592.4
(-22)
22
4748.2
4717.8
23
4880
±
20 4870.5
9
4844.4
36
24 4880
±
20
5000
±
15 4993.4
7
4972.1
-92
28
25 5000
±
15
5115
±
15 5117.0
-2
5100.8
-101
14
26 5115
±
15
5240
±
15 5241.2
-1
5230.3
-115
10
27 5240
±
15
5360
±
15 5366.1
-6
5360.4
-120
-0
28 5360
±
15
5490
±
15 5491.8
-2
5491.1
-131
-1
29 5490
±
15
5615
±
15 5618.3
-3
5622.2
-132
-7
30 5615
±
15
5745
±
15 5745.5
-0
5753.5
-138
-8
31 5745
±
15
5870
±
15 5873.5
-4
5884.9
-140
-15
32 5870
±
15
6000
±
15 6002.3
-2
6016.6
-147
-17
33 6000
±
15
6135
±
15 6132.0
3
6148.6
-149
-14
34 6135
±
15
6265
±
15 6262.3
3
6280.6
-146
-16
35 6265
±
15
6400
±
15 6393.5
7
6412.3
-147
-12
36 6400
±
15
6530
±
15 6525.3
5
6543.8
-144
-14
37 6530
±
15
6660
±
20 6657.9
2
6675.1
-145
-15
38 6660
±
20
6790
±
20 6791.1
-1
6806.0
-146
-16
39 6790
±
20
6920
±
20 6925.0
-5
6936.2
-146
-16
40 6920
±
20
7060
±
20 7059.4
1
7066.0
-146
-6
41 7060
±
20
7190
±
20 7194.5
-4
7195.1
-135
-5
42 7190
±
20
7320
±
20 7330.0
-10
7323.3
-133
-3
43 7320
±
20
7466.1
7450.8
-131
S5. EXPERIMENTAL UNCERTAINTY OF THE
POTENTIAL ENERGY CURVE
To estimate the overall uncertainty in the poten-
tial energy curve, as shown by the blue-shaded area
in Figure 2 (a), we tried out various assignments and
sampled different fits that match these assignments
equally well. Similar to Casey and Leopold, we found
that the quantum number of the vibrational ener-
gies above
4880 cm
−
1
can be assigned by starting at
one of the following values
v
prog
= 21
,
22
,
23
,
24
, and
v
prog
= 25
. For all these five assignments PECs can
be found with matching frequencies. Figure S2 shows
examples of possible curves with different
v
prog
that
all lead to excellent agreement with the experimental
vibrational energy levels (max. error of
14 cm
−
1
, well
within the average experimental uncertainty). Note
that many other fits are possible, as indicated by the
difference between the
v
prog
= 23
curve (blue line)
and our final new fit (dashed purple line), which also
6
is based on
v
prog
= 23
but overall has a smaller er-
ror. We note that we did not include the experimen-
tal isotope shift of some of the vibrational levels ob-
tained from Casey and Leopold as an additional mea-
sure of quality in this procedure, because the isotope
shift itself is only given with large error (
24
±
8 cm
−
1
)
and the final PEC obtained by Casey and Leopold
also disagrees with this (average shift of
36 cm
−
1
).
For these five assignments and for the final assign-
ment from Casey and Leopold that includes the two
4290
/
4570 cm
−
1
levels we drew
7
·
10
5
random sam-
ples of the experimentally observed frequencies within
their experimental uncertainty. Based on these sam-
pled frequencies, we used the RKR method to obtain
a first guess of the experimental PEC. We then refined
that guess using the EMO potential. Finally, from all
these
6
·
7
·
10
5
≈
4
·
10
6
PECs we selected those that
match all experimentally observed frequencies within
their experimental uncertainty. At each bond length,
we then used the extrema of these set of PECs as an
estimate of the experimental uncertainty in the po-
tential energy curve shown as the blue-shaded area in
Figure 2 (a).
TABLE S5: Measured and simulated spectroscopic constants for the
X
1
Σ
+
g
state of Cr
2
. The experimental values of
D
e
assume a vibrational zero-point energy of
0
.
029 eV
. The composite method used in this work gave a zero-point energy
of
0
.
030 eV
. If the original literature did not contain all constants, we computed the constants and show them in square
brackets. For
ω
e
, we use a fit to a second-order Dunham expansion of the lowest 9 vibrational levels. For comparison,
we show the so-obtained
ω
e
also if the original data is available. Note that the original values in Ref. 22 use a different
PEC than the recomputed ones, because the used PEC is not displayed. The data is used for Figure 3 in the main text.
Note that the experimental value of
D
e
(
D
0
) from Ref. 40 may be too large, as the authors assumed that the
2
Σ
+
g
state
is the ground state of Cr
+
2
. Recent experiments indicate that the configuration of the ground state of Cr
+
2
is
12
Σ
+
u
.
41
Similarly, the value of
D
e
(
D
0
) from Ref. 42 may be too small, as the analysis did not take into account the contributions
of electronically excited states.
method
r
e
/
Å
D
e
/
eV
ω
e
/
cm
−
1
∆
G
1
/
2
/
cm
−
1
this work
composite method
1.685
1.58
±
0.02 495
468
MPS-REPT2
1.700
1.52
477
458
other work
semiempirical
43
2.5
1.05
extended Hückel
44
1.7
1.6
SCF-X
α
-SW
44
1.9
GVB-vdW
45
3.0
±
0.6 0.3
±
0.05
110 [106]
108
MGVB
46
1.61
1.86
[600]
[547]
CCSD(T)
47
1.604
766
CCSD(T)
48
1.59
0.38
UCCSD(T)
48
2.54
0.89
[174]
[174]
UBLYP
48
1.7
1.99
[415]
[400]
CASPT2
49
1.71
1.583
625 [591]
[574]
CASPT2
50
1.686
1.538
550 [544]
535
SP-B3P86
51
1.59
1.38
[502]
[427]
MRACPF
52
1.73
1.12
318 [358]
[324]
MRACPF
53
1.72
1.09
339 [148]
[235]
sc-NEVPT2
54
1.662
1.493
575 [560]
[552]
CASPT2
55
1.662
1.618
413 [445]
[410]
MRCIPT2+Q
56
1.756
1.18
332 [344]
[311]
MRCI+Q
56
1.666
1.07
512
MRAQCC
20
1.685
1.355
459 [423]
431
RASPT2
57
1.687
1.6
516
MPS-CASPT2
22
1.681
1.61
480 [516]
[489]
SplitGAS
58
1.671
1.56
484.8
DMRG-ec-MRCI+Q
59
1.71
1.62
477 [496]
[482]
iSUPT2
60
[1.59]
[1.35]
[681]
[456]
FP-AFQMC
61
1.65
±
0.02 1.63
±
0.05 552
±
93 [541] [478]
GASPT2
62
1.67
1.42
RASPT2
21
1.666
1.89
489 [468]
[464]
MPS-NEVPT2
23
1.656
1.43
470 [435]
[422]
SHCI+UCCSD(T)
12
[1.683] [1.551]
[448]
[428]
experiment
1.71
63
1.56
±
0.3
64
1.6788
39
1.472
±
0.06
42
480.6
±
0.5
24
455
±
10
24
1.45
±
0.1
65
470
39
452.34
±
0.02
39
1.56
±
0.06
40
7
−
1.6
−
1.4
−
1.2
−
1
−
0.8
−
0.6
−
0.4
−
0.2
1.6
1.8
2
2.2
2.4
2.6
2.8
3
3.2
binding energy/eV
bond length/Å
v
prog
= 21
v
prog
= 22
v
prog
= 23
v
prog
= 24
̃
v
prog
= 24
v
prog
= 25
Casey/Leopold: Expt. fit
This work: Expt. new fit
FIG. S2. Examples of various expanded Morse oscillator PEC fits to the experimental vibrational data.
v
prog
denotes
the start of the cluster of 20 vibrational levels.
̃
v
prog
denotes that the additional vibrational levels at
4290 cm
−
1
and
4570 cm
−
1
are included in the fit as
v
= 18
and
v
= 21
, respectively.
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•
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•
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•
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•
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•
1994b/1994c Ref. 48
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•
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•
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•
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•
2001 Ref. 67
•
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•
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•
2009 Ref. 20
•
2011 Ref. 22
•
2015 Ref. 61
•
2016a Ref. 21
•
2016b Ref. 23
•
2018 Ref. 59
•
2019 Ref. 60
•
2020 Ref. 12
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