of 6
The Chromium
Dimer:
Closing
a Chapter
of Quantum
Chemistry
Henrik
R. Larsson,
*
Huanchen
Zhai,
C. J. Umrigar,
and Garnet
Kin-Lic
Chan
*
Cite This:
J. Am. Chem.
Soc.
2022,
144, 15932−15937
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*
Supporting
Information
ABSTRACT:
The
complex
electronic
structure
and unusual
potential
energy
curve
of the chromium
dimer
have
fascinated
scientists
for decades,
with
agreement
between
theory
and experiment
so far elusive.
Here,
we present
a new ab initio
simulation
of
the potential
energy
curve
and vibrational
spectrum
that significantly
improves
on all earlier
estimates.
Our data
support
a shift
in
earlier
experimental
assignments
of a cluster
of vibrational
frequencies
by one quantum
number.
The new vibrational
assignment
yields
an experimentally
derived
potential
energy
curve
in quantitative
agreement
with
theory
across
all bond
lengths
and across
all
measured
frequencies.
By solving
this long-standing
problem,
our results
raise
the possibility
of quantitative
quantum
chemical
modeling
of transition
metal
clusters
with
spectroscopic
accuracy.
T
ransition
metal
chemistry
plays
a pivotal
role in catalysis,
biochemistry,
and the energy
sciences,
but the complex
electronic
structure
of the
d
-shells
challenges
our modeling
and
understanding
of such
processes.
1
3
Among
the
most
complicated
of small
transition
metal
molecules
is the
chromium
dimer,
often
described
as a grand
challenge
problem
of small
molecule
quantum
chemistry,
and
whose
unusual
bonding
and
potential
energy
curve
(PEC)
has
puzzled
scientists
for decades.
4
15,21,22
Addressing
this is relevant
for
other
dichromium
compounds,
as well as for other
compounds
where
multiple
metal
metal
bonds
and
spin
coupling
appear.
1
5,7,9,14,16
19
The Cr
Cr
bond
in the bare
dimer
is a
formal
sextuple
bond,
and when
complexed
with
ligands,
was
the first example
of a quintuple
bond.
16,20
Although
the formal
bond
order
is high,
the PEC
inferred
from
photoelectron
spectroscopy
indicates
a short
and weak
bond
with
a narrow
minimum
around
1.68
Å, and an extended
shelf
at around
2.5
Å.
11
The
curve
takes
this form
because
the Cr 4s and 3d
atomic
orbitals
are of very
different
size,
with
the minimum
corresponding
mostly
to 3d orbital
interactions
and the shelf
to
4s orbital
interactions.
Beyond
this
picture,
a quantitative
understanding
remains
lacking.
In particular,
theoretical
predictions
of the binding
curve
deviate
substantially
from
the experimentally
derived
curve,
as well as from
each
other,
while
the experimental
curve
is uncertain
at longer
bond
lengths.
Here
we show
that
a combined
analysis
from
new
numerical
simulations
using
state-of-the-art
quantum
chemistry
methods,
together
with
existing
experimental
data,
yields
a
definitive
picture
of the Cr
2
PEC.
In particular,
our work
suggests
a reassignment
of the vibrational
subbands
in the shelf
region,
bringing
theory
and experiment
finally
into quantitative
agreement.
The complex
electronic
structure
arises
from
the interplay
of
two
types
of electron
correlation.
First,
there
is the spin-
coupling
of the 12 valence
electrons
in the 3d and 4s Cr atoms
shells;
this is termed
static
correlation.
Because
the many-
electron
wave
function
of the valence
electrons
is not well
captured
by a single
determinant,
we refer
to the electronic
structure
as multireference.
Second,
a large
basis
is needed
to
capture
excitations
involving
nonvalence
orbitals;
for example,
the formation
of the 3d
3d
bonds
requires
the 3p electrons
to
move
out of the same
spatial
region
by exciting
to higher
lying
orbitals;
such
effects
are referred
to as dynamic
correlation.
The problem
is computationally
challenging
because
both
the
static
and dynamic
correlation
must
be computed
sufficiently
well
even
for a qualitatively
reasonable
description.
For
example,
the valence
complete
active
space
self-consistent-
field method
(which
treats
the valence
static
correlation
exactly
but
neglects
the dynamic
correlation)
does
not
yield
a
minimum
near
the equilibrium
bond
length,
21
while
the
gold-standard
treatment
of dynamical
correlation,
coupled
cluster
singles,
doubles
and perturbative
triples
(CCSD(T))
also
does
not yield
a reasonable
bond
length,
nor does
it
display
a shelf
region.
22
Figure
1 shows
curves
from
calculations
over
many
years;
the lack of consensus
is striking.
Even
when
limited
to studies
from
the last decade,
there
is a spread
of over
0.6 eV in the
predicted
binding
energy
and 0.2
0.34
eV across
the whole
curve.
Experimentally,
while
multiple
techniques
have
shed
light
on the spectroscopic
constants,
information
on the full
PEC
comes
from
a photoelectron
spectroscopy
study
of
Cr
2
.
10,11
This
measured
29 vibrationally
resolved
transitions,
and by assigning
these
to specific
vibrational
quantum
numbers
v
, a PEC
was derived
using
the Rydberg
Klein
Rees
method.
However,
the assignments
above
v
= 9 are uncertain,
in
particular,
the assignment
of the starting
quantum
number
v
prog
of a high-lying
20 member
vibrational
progression.
Together
with
nonuniqueness
in the PEC
fit, this leads
to considerable
uncertainty
in the experimental
PEC.
This
is shown
by the
Received:
June
16, 2022
Published:
August
24,
2022
Communication
pubs.acs.org/JACS
© 2022
The Authors.
Published
by
American
Chemical
Society
15932
https://doi.org/10.1021/jacs.2c06357
J. Am. Chem.
Soc.
2022,
144, 15932
15937
shaded
region
of Figure
2a, which
shows
the range
of
experimental
PEC
arising
from
different
assignments
(
v
prog
=
21
25),
all of which
match
the observed
vibrational
levels
within
their
experimental
uncertainty;
in the shelf-region
the
uncertainty
is over
0.1 eV (further
details
in the SI).
To compute
a more
accurate
PEC,
we will
employ
a
composite
method
starting
from
the scalar
relativistic
“exact
two-component”
(X2C)
Hamiltonian,
23,24
which
is based
on
two contributions.
The
first contribution
estimates
the exact
chromium
dimer
PEC
in a moderate
basis
(Dunning’s
cc-
pVDZ-DK
basis,
here
dubbed
PDZ,
25
with
a frozen
neon
core,
correlating
28 electrons
in 76 orbitals).
For this,
we use data
from
very
large
ab initio
density
matrix
renormalization
group
(DMRG)
calculations
(using
up to bond
dimension
28000
and
SU(2)
symmetry),
together
with
selected
heat-bath
config-
uration
interaction
(SHCI)
data
computed
earlier
by one of
us.
26
The
second
contribution
targets
the remaining
dynamic
correlation.
For
this,
we use data
computed
from
multi-
Figure
1.
Some
of the simulated
potential
energy
curves
(PECs)
of the chromium
dimer
available
in the literature,
labeled
by year.
The red curve
marks
this work.
The inset
shows
selected
PECs
from
2011
onward.
Figure
2.
Theoretical
and experimental
potential
energy
curves
(PECs)
of the chromium
dimer.
(a) Blue:
experimental
PEC
from
ref 11. The blue-
shaded
area estimates
the uncertainty
from
the experimental
PEC
fit. Purple:
new experimental
PEC
from
vibrational
assignment
in this work.
Red:
computed
PEC
with
error
estimates.
(b) Estimated
simulation
errors.
ε
PDZ
: error
of “exact”
estimate
of the cc-pVDZ-DK
basis
curve.
ε
CBS
: error
in
the complete
basis-set
extrapolation.
ε
REPT
: error
of the dynamic
correlation
correction.
(c) New
theoretical
and experimental
PEC
compared
with
the next
best PEC
in the literature
from
Li et al. (gray
curve).
26
Journal
of the American
Chemical
Society
pubs.acs.org/JACS
Communication
https://doi.org/10.1021/jacs.2c06357
J. Am. Chem.
Soc.
2022,
144, 15932
15937
15933
reference
perturbation
theory
(using
an efficient
formulation
of
the restraining
the excitation-order
Hamiltonian
27,28
within
the
language
of matrix
product
state
perturbation
theory,
28
30
starting
from
the 12 electron,
12 orbital
valence
complete
active
space)
computed
using
cc-pV
N
Z-DK
basis
sets up to
quintuple
zeta,
as well as unrestricted
CCSD(T)
data.
26
This
dynamical
correlation
correction
is then
extrapolated
to the
basis
set limit.
We perform
the simulations
with
the
PYSCF
and
BLOCK
2 program
packages.
30
33
It is important
to estimate
the error
in these
various
contributions.
For
the PDZ
curve,
as DMRG
and
SHCI
provide
independent
extrapolations
to numerical
exactness
with
similar
confidence,
we take the average
of the DMRG
and
SHCI
data
as the curve,
with
half the difference
as the error
(
ε
PDZ
). For the dynamic
correlation
correction,
the CCSD(T)
data is expected
to be less accurate
than
the REPT
data,
due to
the multireference
nature
of the correlation.
Thus,
we only
use
the REPT
data,
and use half the difference
from
CCSD(T)
as
the error
(
ε
REPT
). Finally,
the basis
set error
is estimated
as the
standard
deviation
of the
complete
basis
set (CBS)
extrapolation
fit (
ε
CBS
). Taking
these
three
error
contributions
as independent,
the total
error
is then
the square
root
of the
quadratic
sum.
Note
that it is difficult
to assert
the statistical
significance
of these
error
estimates;
however,
they
provide
a
useful
measure
of accuracy.
Using
the new
theoretical
PEC,
we compute
the detailed
vibrational
spectrum
by solving
the vibrational
Schro
̈
dinger
equation.
We
use
this
to then
reassign
the
measured
experimental
peaks
from
the photoelectron
spectrum,
11
and
with
these
assignments,
solve
the inverse
Schro
̈
dinger
equation
to derive
a new experimental
PEC.
We
show
the
computed
PEC
in Figure
2a.
The
accompanying
error
estimates
for the PDZ
curve,
the dynamic
correlation
correction,
and the CBS
extrapolation
are shown
in
Figure
2b.
ε
PDZ
is quite
small
<0.01
eV,
demonstrating
remarkable
agreement
in the “exact”
PDZ
energies
from
DMRG
and SHCI.
ε
CBS
is also small
<0.012
eV. The
largest
error
is from
the dynamic
correlation
which
is as large
as
0.034
eV at
2.0
2.25
Å. As discussed
in the SI, this likely
reflects
the poor
performance
of the CCSD(T)
method
used
to estimate
the error,
and thus this large
error
is a conservative
estimate
(other
ways
of estimation
given
in the SI). Overall,
we
find good
agreement
with
the existing
experimental
PEC,
and
one
that
is significantly
improved
over
all previous
computations
in the literature
(the
next
best
match
is shown
in Figure
2c, which
has substantial
disagreement
in the shelf
region).
Unlike
some
earlier
predictions,
no double
minimum
is observed.
The
largest
uncertainties
in the theoretical
curve
lie outside
of the region
of the PEC
with
large
experimental
uncertainty;
we return
to this point
in the following.
Figure
3 shows
spectroscopic
constants
derived
from
the
current
and earlier
PECs,
compared
to experiment.
We find
very
good
agreement
with
experiment;
the improvement
in
theoretical
predictions
over
time
is shown
in the lowest
panel.
Note,
however,
that the spectroscopic
constants
only
measure
the quality
of the PEC
only
near
the minimum.
In fact,
the
other
studies
with
spectroscopic
constants
with
small
mean
error
are associated
with
PECs
of poor
overall
shape
(see
Figure
1), reflecting
a poor
description
of the full vibrational
spectrum.
We now more
carefully
examine
the shelf
region
of the PEC,
where
there
is the largest
deviation
from
the experimentally
Figure
3.
Simulated
spectroscopic
constants
of Cr
2
over
time:
r
e
(equilibrium
bond
length),
D
e
(well-depth),
ω
e
(harmonic
frequency)
and
Δ
G
1/2
(fundamental
frequency).
Blue:
most
recent
experimental
result;
shaded
area:
experimental
uncertainty
(that
of
r
e
is not reported).
The
lowest
panel
shows
the evolution
of the average
absolute
percentage
error
(in case all four constants
are available;
error
bar shows
min/max
error).
Further
data
is shown
in the SI (Table
S5).
Journal
of the American
Chemical
Society
pubs.acs.org/JACS
Communication
https://doi.org/10.1021/jacs.2c06357
J. Am. Chem.
Soc.
2022,
144, 15932
15937
15934
derived
curve.
Ref 11 contains
a progression
of 20 vibrational
levels
starting
from
4880
cm
1
, with
a spacing
of approximately
128
cm
1
. Casey
and
Leopold
tried
various
assignments,
ultimately
assigning
the first frequency
of this cluster
to
v
prog
=
24. The
vibrational
frequencies
for our theoretical
PEC
are
shown
in Figure
4a. With
the old assignment
of
v
prog
, our
simulated
frequencies
of this cluster
consistently
disagree
with
the experimental
result
by approximately
a single
energy
quantum
in the shoulder
region
of the PEC
(Figure
4b). This
is surprising
given
the small
theoretical
estimate
of the error
in
this region,
and suggests
that
we should
simply
change
the
assignment
from
v
prog
= 24 to
v
prog
= 23. This
reduces
the root-
mean-square
deviation
(RMSD)
from
113 to 19 cm
1
. The
largest
discrepancy
now occurs
for one of the lower
states
(
v
=
7), consistent
with
the region
of largest
uncertainty
in the
theoretical
calculations,
between
2.0
2.25
Å, (see
Figure
4b).
Notably,
the experimental
vibrational
frequencies
have
a
RMSD
of 16 cm
1
, similar
to the RMSD
of the computed
frequencies
with
the new assignment
(see Table
S4 in the SI).
While
theory
does
not allow
for a statistical
estimate
of
certainty,
the quantitative
agreement
between
the theoretical
and experimental
vibrational
frequencies
across
the measured
peaks
is striking
and is our main
result.
As a consequence,
solving
the inverse
Schro
̈
dinger
equation
with
the suggested
new
assignment
of
v
prog
leads
to a revised
estimate
of the
experimental
PEC
shown
in Figure
2a. The
revised
PEC
demonstrates
an excellent
match
between
theory
and experi-
ment.
The computational
prediction
of the ground-state
PEC
of a
diatomic
that
is quantitatively
consistent
with
experiment
might
seem
to be a standard
task,
but in the case
of the
chromium
dimer
it has been
a challenge
for decades.
Our work
shows
that this goal can finally
be achieved;
as one metric,
the
average
error
in the vibrational
spectrum
computed
from
the
theoretical
PEC
is now
comparable
to the average
uncertainty
of the vibrational
peaks
measured
in experiment.
While
this
arguably
brings
to a close
a storied
problem
of computational
quantum
chemistry,
it opens
the door
to many
others,
in
particular,
the applications
of the
theoretical
techniques
discussed
here
not only
to other
complex
multiple
metal
metal
bonded
species,
but more
generally
to the quantitative
spectroscopic
modeling
of transition
metal
clusters.
Figure
4.
Vibrational
ladder.
(a) Experimental
(left)
vibrational
frequencies
compared
with
the simulated
frequencies
(right).
The experimental
data lacks
values
between
3250
and
4750
cm
1
; the original
assignment
of frequencies
with
quantum
numbers
larger
than
9 is not fully
certain.
(Note
that the two measured
frequencies
labeled
in brackets
have
not definitively
been
associated
with
the Cr
2
ground
state,
see ref 11.)
The
simulated
frequencies
enable
a new
assignment,
shown
in red. (b) Vibrational
error
of the new
(old)
assignment
shown
as positive
(negative)
values,
as a function
of the right
classical
turning
point
of each
state
(based
on our new experimental
PEC).
For the 9 lowest
vibrational
levels,
assiociated
with
turning
points
up to 2.1 Å, the assignment
does
not change
and the errors
are identical.
For the 20 higher
lying
levels,
the old
(new)
assignment
uses
v
prog
= 24(23).
Journal
of the American
Chemical
Society
pubs.acs.org/JACS
Communication
https://doi.org/10.1021/jacs.2c06357
J. Am. Chem.
Soc.
2022,
144, 15932
15937
15935
ASSOCIATED
CONTENT
Data Availability
Statement
Data
for reproducing
Figure
2a is available
at https://github.
com/h-larsson/Cr2Pes22.
Further
data
is available
from
the
authors
upon
reasonable
request.
*
Supporting
Information
The
Supporting
Information
is available
free
of charge
at
https://pubs.acs.org/doi/10.1021/jacs.2c06357.
Electronic
structure
methods
(Section
S1),
vibrational
spectrum
(S2),
PEC
fit (S3),
vibrational
assignment
(S4),
uncertainty
estimate
of the experimental
PEC
(S5),
as well as references
for Figure
1 (S6)
(PDF)
AUTHOR
INFORMATION
Corresponding
Authors
Henrik
R. Larsson
Division
of Chemistry
and Chemical
Engineering,
California
Institute
of Technology,
Pasadena,
California
91125,
United
States; Department
of Chemistry
and Biochemistry,
University
of California
Merced,
Merced,
California
95343,
United
States;
orcid.org/0000-0002-
9417-1518;
Email:
larsson22_cr2@larsson-research.de
Garnet
Kin-Lic
Chan
Division
of Chemistry
and Chemical
Engineering,
California
Institute
of Technology,
Pasadena,
California
91125,
United
States
;
Email:
garnetc@
caltech.edu
Authors
Huanchen
Zhai
Division
of Chemistry
and Chemical
Engineering,
California
Institute
of Technology,
Pasadena,
California
91125,
United
States;
orcid.org/0000-0003-
0086-0388
C. J. Umrigar
Laboratory
of Atomic
and Solid State Physics,
Cornell
University,
Ithaca,
New York 14853,
United
States
Complete
contact
information
is available
at:
https://pubs.acs.org/10.1021/jacs.2c06357
Notes
The
authors
declare
the
following
competing
financial
interest(s):
G.K.C.
is part-owner
of QSimulate,
Inc.
ACKNOWLEDGMENTS
We thank
Doreen
Leopold
and
Alec
Wodtke
for helpful
discussions.
Work
by G.K.C.
was
supported
by the US
National
Science
Foundation
(NSF)
via grant
no. CHE-
2102505.
G.K.C.
acknowledges
additional
support
from
the
Simons
Foundation
via the Many-Electron
Collaboration
and
the Investigator
Award.
Work
by H.R.L.,
H.Z.,
and C.J.U.
was
supported
by the Air Force
Office
of Scientific
Research,
under
Award
FA9550-18-1-0095.
H.R.L.
acknowledges
support
from
a postdoctoral
fellowship
from
the
German
Research
Foundation
(DFG)
via grant
LA 4442/1-1
during
the first
part of this work.
Some
of the computations
were
conducted
at
the Resnick
High
Performance
Computing
Center,
a facility
supported
by the Resnick
Sustainability
Institute
at the
California
Institute
of Technology.
The
SHCI
calculations
were
performed
on the Bridges
computer
at the Pittsburgh
Supercomputing
Center
under
grant
PHY170037.
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