# Combining complex conjugation, time-reversal, and spin symmetry projection
of coupled cluster wave functions

Ruiheng Song<EMAIL_ADDRESS>Department of Chemistry, Rice University, Houston,
TX 77005-1892  Thomas M. Henderson  Department of Chemistry, Rice University,
Houston, TX 77005-1892  Department of Physics and Astronomy, Rice University,
Houston, TX 77005-1892  Gustavo E. Scuseria  Department of Chemistry, Rice
University, Houston, TX 77005-1892  Department of Physics and Astronomy, Rice
University, Houston, TX 77005-1892

###### Abstract

Complex conjugation symmetry breaking and restoration generate two non-
orthogonal configurations at the Hartree-Fock level that can capture static
correlation naturally. In conjunction with broken spin-symmetry coupled
cluster theory, the symmetry-projected wave function shows good agreement with
full configuration interaction in beryllium hydride insertion, lithium
fluoride dissociation, and symmetric stretching of tetrahedral H4. By adding
spin flip projection, we can also recover time reversal symmetry in the same
coupled cluster framework.

##  I Introduction

The wave functions we use in quantum chemistry are typically real-valued. That
is, the molecular orbital coefficients in Hartree-Fock or density functional
theory and the wave function amplitudes in methods such as coupled cluster
theory are generally chosen to be real. Choosing to work with real quantities
simplifies implementation, since we do not need to keep track of complex
conjugates, and also reduces storage requirements. More precisely, these wave
functions possess complex conjugation symmetry, which is essentially a
realization of the fact that in a basis in which the Hamiltonian matrix is
real, its eigenvectors can be chosen real. We note that when we use a plane
wave basis, the basis itself is not real and consequently neither are the
orbital coefficients, but this does not imply that calculations in a plane
wave basis necessarily break complex conjugation symmetry.

On the other hand, while complex wave functions are infrequently used in
atomic orbital-based quantum chemistry, it has long been known that one can
break complex conjugation symmetry at the mean-field level to obtain, in
principle, variationally superior energies.[1, 2, 3] Indeed, research in the
past several years has emphasized that a complex conjugation-broken wave
function has a more flexible form in solving electronic structure problems.
For example, many complex determinants smoothly connect different
solutions.[4, 5] Holographic Hartree-Fock provides continuous solutions across
all geometries.[6] Complex conjugation symmetry breaking is particularly
important for general Hartree-Fock (GHF) wave functions which break both
$S^{2}$ and $S_{z}$ symmetry since it is only on breaking complex conjugation
symmetry that GHF permits a description of non-coplanar magnetism.[7]

The fact remains, however, that when complex conjugation is a symmetry of the
Hamiltonian, the wave function in a real basis should be real or, more
precisely, should be an eigenstate of the complex conjugation operator $K$:

$K|\Psi\rangle=e^{\mathrm{i}\theta}|\Psi\rangle$ (1)

where the phase $\mathrm{e}^{\mathrm{i}\,\theta}$ can always be chosen to be
$\pm 1$ by multiplying $|\Psi\rangle$ by the appropriate and physically
inconsequential global phase factor.

Previous work has established many symmetry-projected wave function methods
including Hartree-Fock,[8, 9] configuration interaction,[10] coupled cluster
theory,[11, 12, 13, 14] and perturbation theory.[15] Most of this work has
focused on the projection of spin symmetry, although the projection of number
symmetry is frequently used in nuclear physics.[16] Complex conjugation
symmetry is a somewhat different beast. It is a discrete symmetry, and because
the complex conjugation operator is anti-unitary it does not have an
observable quantum number. The implementation of complex conjugation
projection with post-HF methods remains unexplored.

In this work, we review the complex conjugation operator and implement the
complex conjugation projection of Hartree-Fock and coupled cluster theory. In
addition, we add spin flip projection to recover time-reversal symmetry. We
test these methods on small molecules with complex restricted Hartree-Fock,
unrestricted Hartree-Fock (UHF), and generalized Hartree-Fock, as well as with
coupled cluster with single and double excitations (CCSD) based on all of
these mean-field references. We aim to show that complex conjugation
projection can be a useful tool for multi-reference problems. We also show a
few pitfalls which might arise, since frequently projecting one symmetry also
projects or at least approximately projects others, and we show how we can
circumvent these difficulties.

##  II Background

Before we can discuss complex conjugation projection, we must briefly review
the complex conjugation and time reversal operators. Readers familiar with
these ideas may skip the following subsections, though we use them to
establish our notation.

###  II.1 The Complex Conjugation Operator

The complex conjugation operator is an anti-unitary operator which
satisfies[17]

$\displaystyle\langle K\,\psi|\phi\rangle$
$\displaystyle=\langle\psi|K^{+}\,\phi\rangle^{*},$ (2) $\displaystyle
K^{+}\,K$ $\displaystyle=K\,K=1.$ (3)

Here, we use $K^{+}$ for the adjoint of the complex conjugation operator to
distinguish it from the Hermitian adjoint. Note that the standard bra-ket
notation is not particularly well equipped to handle antiunitary operators
like $K$. A symbol such as $\langle\phi|K|\psi\rangle$ is somewhat ambiguous,
as one could understand it to mean acting $K$ to the right or the left.
Generally, these two operations will yield different results, so when there is
any risk of ambiguity we will include $K$ inside the bra or ket, as we have
done in Eqn. 2.

When acting on a complex number, $K$ is just a simple complex conjugation.
However, its action on a quantum state is slightly different. This is perhaps
easiest to see by way of an example. Suppose that $|\Psi\rangle$ is a single
determinant with real atomic orbitals and with complex molecular orbital
coefficients $\bm{C}$. Then $K|\Psi\rangle$ is also a single determinant, but
its molecular orbital coefficients are $\bm{C}^{\star}$:

$K|\Psi(\bm{C})\rangle=|\Psi(\bm{C}^{\star})\rangle.$ (4)

We can always find a unitary transformation $R_{K}$ to do the same, so that we
can write

$K|\Psi\rangle=R_{K}|\Psi\rangle.$ (5)

We will use the operator in this form in our projected wave function methods.

###  II.2 The Time Reversal and Spin Flip Operators

We introduce the spin flip operator $F=e^{i\pi S_{y}}$ and time reversal
operator $\mathcal{T}=FK$. Together with the identity operator and complex
conjugation operator, the operator product is closed:

$\displaystyle K\,F$ $\displaystyle=F\,K$ $\displaystyle=\mathcal{T},$ (6a)
$\displaystyle K\,\mathcal{T}$ $\displaystyle=\mathcal{T}\,K$
$\displaystyle=F,$ (6b) $\displaystyle F\,\mathcal{T}$
$\displaystyle=\mathcal{T}\,F$ $\displaystyle=-K.$ (6c)

While acting $K$ twice does nothing, acting $F$ or $\mathcal{T}$ on a state
twice returns the same state up to a sign that is positive if the state has an
even number of particles and negative if it does not. This is because acting
$F$ twice rotates each spin by $360\degree$ which is equivalent to multiplying
by $-1$.

Because $K$, $F$, and $\mathcal{T}$ are not independent symmetries, we can
project any two of them and the third is also automatically projected. We take
advantage of this fact below to project complex conjugation and time reversal,
by projecting complex conjugation and spin flip. We should also note that spin
flip projection is equivalent to half-spin projection,[18, 19] which
eliminates half of the spin contaminants from an unrestricted wave function.
More specifically, it eliminates contaminants whose spin parity differs from
that of the target. Half-spin projection onto a singlet ($S=0$), for example,
eliminates contaminants from triplet, septet, and so on (i.e. those with
$S=2\,n+1$).

##  III Complex Conjugation Projection

###  III.1 Complex Conjugation Projected Hartree-Fock

Starting from a state $|\Psi\rangle$ which breaks complex conjugation
symmetry, we can form a basis $\\{|\Psi\rangle,K\,|\Psi\rangle\\}$, and we can
diagonalize the Hamiltonian in this basis. Since the Hamiltonian commutes with
$K$, the resulting eigenvectors will naturally be eigenvectors of $K$. In this
sense, we can construct the complex conjugation projection operator as

$P_{K}=1+\mathrm{e}^{\mathrm{i}\,\theta}\,K.$ (7)

It is easy to show that that

$K\,P_{K}|\Psi\rangle=\mathrm{e}^{-\mathrm{i}\,\theta}\,P_{K}|\Psi\rangle,$
(8)

so that the projected state is an eigenfunction of $K$.

The projection operators for spin and point group are Hermitian. The complex
conjugation projector in general is not, so we must be careful to distinguish
between the complex conjugation projector acting to the right and to the left.
We define $\overleftarrow{P_{K}}$ to mean the projection operator acting to
the left. Given this notation, we can define the complex conjugation projected
Hartree-Fock (KHF) energy as

$E[Z^{\dagger},Z]=\frac{\langle\Phi|e^{Z^{\dagger}}\,\overleftarrow{P_{K}}\,H\,P_{K}\,e^{Z}|\Phi\rangle}{\langle\Phi|e^{Z^{\dagger}}\,\overleftarrow{P_{K}}\,P_{K}\,e^{Z}|\Phi\rangle}$
(9)

where the reference $|\Phi\rangle$ is chosen to be real and the complex-valued
Thouless transformation[20] $Z$, given by

$Z=\sum_{i,a}z_{i}^{a}\,c_{a}^{\dagger}\,c_{i},$ (10)

is used to optimize the broken-symmetry reference. Here, $i$ and $a$ are
occupied and unoccupied orbitals, respectively. We will also need to solve a
$2\times 2$ CI problem to determine the coefficient
$f_{K}=\mathrm{e}^{\mathrm{i}\,\theta}$ for each pair of $[Z^{\dagger},Z]$.

As we noted earlier, we can project both complex conjugation and time reversal
(another anti-unitary symmetry) by instead projecting complex conjugation and
spin-flip symmetries. This means writing

$E[Z^{\dagger},Z]=\frac{\langle\Phi|e^{Z^{\dagger}}\,\overleftarrow{P_{K}}\,P^{\dagger}_{F}\,H\,P_{F}\,P_{K}\,e^{Z}|\Phi\rangle}{\langle\Phi|e^{Z^{\dagger}}\,\overleftarrow{P_{K}}\,P^{\dagger}_{F}\,P_{F}\,P_{K}\,e^{Z}|\Phi\rangle}.$
(11)

Here $P_{F}=1\pm F$, where the sign is $1$ for even spin (e.g. a ground-state
singlet) and $-1$ for odd spin (e.g. a triplet). We optimize this energy
functional with the BFGS algorithm; the analytical gradient is included in the
appendix.

Following the work of Ghassemi-Tabrizi and coworkers[21], we could eliminate
the phase in the projection operator by incorporating a phase into the
optimized reference $|\tilde{\Phi}\rangle=e^{Z}|\Phi\rangle$. That is, if we
write

$|0\rangle=\mathrm{e}^{-\mathrm{i}\,\theta/2}\,|\tilde{\Phi}\rangle,$ (12)

then we have

$\displaystyle P_{K}|\tilde{\Phi}\rangle$
$\displaystyle=P_{K}\mathrm{e}^{\mathrm{i}\,\theta/2}\,\mathrm{e}^{-\mathrm{i}\,\theta/2}|\tilde{\Phi}\rangle$
$\displaystyle=P_{K}\mathrm{e}^{\mathrm{i}\,\theta/2}|0\rangle$
$\displaystyle=\left(1+\mathrm{e}^{\mathrm{i}\,\theta}K\right)\mathrm{e}^{\mathrm{i}\,\theta/2}|0\rangle$
$\displaystyle=\mathrm{e}^{\mathrm{i}\,\theta/2}|0\rangle+\mathrm{e}^{\mathrm{i}\,\theta}\,\mathrm{e}^{-\mathrm{i}\,\theta/2}\,K|0\rangle$
$\displaystyle=\mathrm{e}^{\mathrm{i}\,\theta/2}\left(1+K\right)|0\rangle=\mathrm{e}^{\mathrm{i}\,\theta/2}\,P_{K}|0\rangle.$
(13)

In other words, given a complex determinant $|\tilde{\Phi}\rangle$ which is
projected by $P_{K}=1+\mathrm{e}^{\mathrm{i}\,\theta}K$ so that the projected
state $P_{K}|\tilde{\Phi}\rangle$ is a $K$ eigenstate with eigenvalue
$\mathrm{e}^{-\mathrm{i}\,\theta}$, we can construct an associated complex
determinant $|0\rangle$ such that the projector is simply $1+K$ and the
corresponding $K$ eigenvalue of $P_{K}|0\rangle$ is just 1.

###  III.2 Connection with Other Multireference Methods

Before we discuss complex conjugation projected coupled cluster theory, let us
take a moment to consider the structure of the KHF state. To simplify the
presentation, we assume that we are projecting a complex RHF determinant that
has only one (complex) occupied orbital $|\chi\rangle$. We can write $\chi$ in
terms of two two real molecular orbitals $\phi_{p}$ and $\phi_{q}$ as

$|\chi\rangle=\cos(\theta)\,|\phi_{p}\rangle+\mathrm{i}\sin(\theta)\,|\phi_{q}\rangle$
(14)

where $\theta$ is a rotation angle. The determinant is then

$\displaystyle|\chi_{\uparrow}\chi_{\downarrow}\rangle$
$\displaystyle=\cos^{2}(\theta)|\phi_{p,\uparrow}\,\phi_{p,\downarrow}\rangle-\sin^{2}(\theta)|\phi_{q,\uparrow}\,\phi_{q,\downarrow}\rangle$
(15) $\displaystyle+\textrm{imaginary part}.$

After projection, only the real part survives, and it takes the form of a
perfect pairing generalized valence bond state, except that $\phi_{p}$ and
$\phi_{q}$ need not be individually orthonormal (but can be).

Complex conjugation projected Hartree-Fock is a simple non-orthogonal
configuration interaction (NOCI). As such, it has the potential to describe
some nearly degenerate systems. Because it is parametrized by a single complex
determinant $\mathrm{e}^{Z}|\Psi\rangle$, it is easy to optimize the complex
orbitals defined by the Thouless parameters $Z$, avoiding any arbitrary
choices one might otherwise have to make in the NOCI. The price to pay is that
complex conjugation projection, on its own, involves only two determinants.
Fortunately, it can be combined with spin and point group symmetry projection
to obtain a much more complete description of the strong correlations in a
molecule while still depending on only one Thouless operator $Z$.

###  III.3 Complex Conjugation Projected Coupled Cluster

After minimizing the PHF energy to get the KHF broken symmetry determinant
$|\tilde{\Phi}\rangle=e^{Z}|\Phi\rangle$, we can use the procedure of Eqn. 13
to reduce the projection operator to simply $1+K$. We discuss the reason for
this gauge-fixing step below. We can then add coupled cluster theory to
recover the dynamic correlation. We mimic the traditional coupled cluster
equations to write down projected coupled cluster equations

$\displaystyle E$ $\displaystyle=\langle 0|H\,P_{F}\,P_{K}\,e^{T}|0\rangle$
(16) $\displaystyle 0$
$\displaystyle=\langle\mu|(H-E)\,P_{F}\,P_{K}\,e^{T}|0\rangle$ (17)

Here $\mu$ denotes excited states of the broken symmetry reference, and the
cluster operator T creates excitations out of $|0\rangle$. The key step in
solving the projected coupled cluster equations is evaluating

$\langle 0|K\,H\,e^{T}|\Phi\rangle=\langle 0|R_{K}\,H^{*}\,e^{T^{*}}|0\rangle$
(18)

where a $*$ on an operator means taking the complex conjugate of its
coefficients. For example, $T^{\star}$ has the usual form of the cluster
operator but uses the complex conjugate of the cluster amplitudes.

By using the Thouless theorem and replacing $P_{K}$ with
$\overleftarrow{P_{K}}$ so that we may act the projection operator to the
left, we have

$\langle 0|R_{K}=\langle 0|R_{K}|0\rangle\langle 0|e^{V}$ (19)

where $V$ is a pure 1-body de-excitation operator. Then Eq. 18 becomes

$\displaystyle\langle 0|R_{K}\,H^{*}\,e^{T^{*}}|0\rangle=\langle
0|R_{K}|0\rangle\langle 0|\,H_{V}^{*}\,e^{V}e^{T^{*}}|0\rangle,$ (20)
$\displaystyle H_{V}^{*}=e^{V}H^{*}e^{-V}.$ (21)

One can write

$\mathrm{e}^{V}\,\mathrm{e}^{T^{\star}}|0\rangle=\mathrm{e}^{W}|0\rangle$ (22)

where $W$ is a pure excitation operator. We refer to this approach as the
disentangled cluster approximation,[11] although we note that the
approximation is not in writing the product of exponentials in terms of a
single exponential excitation operator $W$ but in truncating $W$ (which
consists of a normalization constant and all possible excitations) to a lower
operator rank. We have considered various approaches to extract an approximate
$W$ from $V$ and $T$,[11, 13] and here we just match the power series
expansion of $\mathrm{e}^{W}|0\rangle$ in terms of $W$ to that of
$\mathrm{e}^{V}\mathrm{e}^{T^{\star}}|0\rangle$ in terms of $T^{\star}$. For
more details consult Ref. 13. We note that the effects of single excitations
$T_{1}$ can be evaluated without approximation.

##  IV Results

We test the proposed methods on small molecules with the cc-pVDZ basis set.
Note that generally complex conjugation symmetry does not spontaneously break
in Hartree-Fock, but in KHF it always does so by optimizing the mean-field
state in the presence of the projection operator in what is known as the
variation after projection approach. For projected coupled cluster theory, we
consider only CCSD (that is, $T$ is limited to single- and double-excitations)
and we approximate the disentangled cluster operator $W$ to likewise consist
only of a constant plus single- and double-excitations. We match the two power
series through second order in $W$ and $T_{2}$. It would be interesting to
optimize orbitals not for PHF but for PCC as well, but this requires extra
effort and is beyond the scope of this work.

Figure 1: Top panel: Energy of various HF and CCSD in LiF dissociation. Bottom
panel: Energy error in LiF dissociation.

###  IV.1 LiF Dissocation

We start with the dissociation of LiF, which is a well-known multi-reference
problem[22, 23] The molecule shows ionic character at short bond lengths, but
dissociates into neutral atoms. Both configurations have ${}^{1}\Sigma_{+}$
symmetry, causing challenges in actual calculation.

In this case, we allow $S^{2}$ symmetry to break to account for the strong
correlations between the two neutral atoms and add complex conjugation
projection (but not spin symmetry projection).

Figure 1 shows the LiF dissociation curve calculated with several methods: HF
and KHF, CCSD and KCCSD, and the full configuration interaction (FCI)
reference.

Let us begin with the HF and KHF data. The ionic RHF and covalent UHF
solutions cross at a bond length of around 3.0 Å. We can follow the UHF
solution only slightly past the crossing point. When applying complex
conjugation to UHF to get KUHF, the energy curve evolves to the correct limit
at a large bond length and avoids the derivative discontinuity in the middle.
Although KUHF looks much better than RHF or UHF, the Coulson–Fischer point at
which spin symmetry breaks is still perceptible (see Fig. 2), although the
point of spin symmetry breaking is extended from roughly 3 Å to about 3.5 Å.

Now we turn to the CCSD results. In this case, we use the RHF reference for
smaller bond lengths and the UHF reference for larger bond lengths. The fact
that the reference changes explains the discontinuity observed in the CCSD
curve in the bottom panel of Fig. 1. For shorter bond lengths, KCCSD is much
more accurate than CCSD, but the energy error increases significantly once
spin symmetry is broken, and unrestricted CCSD and KCCSD have similar errors
overall. Importantly, however, where UCCSD is not even continuous, KUCCSD
yields a continuous albeit not smooth dissociation profile.

Figure 2: Expectation value of $S^{2}$ of the broken symmetry reference state
in LiF dissociation.

Figure 3: Top panel: Energy of various HF in symmetric stretching of H4 .
Bottom panel: Energy of various CC in symmetric stretching of H4.

###  IV.2 H4

Tetrahedral H4 is a small molecule with spin frustration. Due to its unique
geometry, one can find a real RHF, a real UHF, a real GHF, and a complex GHF
solution.[7] All mean-field solutions other than real RHF break time-reversal
symmetry, so, in this case, we restore both time reversal and complex
conjugation (by, as noted, projectively restoring complex conjugation and
spin-flip symmetries). In the dissociation, there is a singlet-triplet
crossing at short bond length. Thus, we use $1+F$ for singlet (denoted as S0)
and $1-F$ for triplet (denoted as S1) projection respectively.

Figure 3 shows that at the mean-field level, the system is not bonded. Complex
conjugation projected GHF (KGHF) also has no bond, though complex conjugation
and singlet projected GHF (S0 KGHF) has a very small minimum at an H-H
distance of about 1.9 Bohr. Also, we note that the crossing of singlet and
triplet KGHF occurs for hydrogen-hydrogen bond lengths about 1.2 Bohr, at a
slightly smaller bond length than the FCI singlet and triplet solutions cross.

As a system with only four electrons, the complex GHF-based CCSD (cGCCSD) is
not too far from FCI, but the wave function switched from triplet to singlet
at 1.25 Bohr, where the FCI solutions cross. That is, the cGCCSD tracks the
lower energy FCI solution rather than remaining on a single state. The same
behavior has been seen elsewhere in the GCCSD solution for O2
dissociation.[24] Generally, KGCCSD is much closer to FCI than is GCCSD, but
it also tracks the lower energy FCI solution. Including time reversal
projection not only yields improved energies but also allows us to partially
control the spin state onto which we project. Figure 4 shows the maximum
errors are about 1 and 3 milliHartree for singlet and triplet respectively.
However, at sufficiently short bond length we are unable to fully control spin
symmetry just by complex conjugation and time reversal projection since, as we
noted, this amounts to half-spin projection. Evidently, full spin projection
is needed. We note, though, that projecting both complex conjugation and time
reversal amounts to a PHF which is a linear combination of 4 determinants,
while fully spin-projected GHF requires integrating over three continuous
Euler angles, which would generally be expected to require a much larger
number of states in the PHF.

Figure 4: Energy error of projected CC in symmetric stretching of H4 .

###  IV.3 BeH2

Insertion of beryllium into H2 is another classic example in the
literature.[25, 4, 26] We consider the C2v geometry, where the beryllium atom
is at the origin and the hydrogen atoms have coordinates
$\big{(}x,\pm\left(1.344-0.46\,x\right),0\big{)}$ where the units are in Å.

The strong correlations arise from the crossing of the frontier A1 and B2
molecular orbitals as $x$ evolves from 0 (the linear geometry) to 2. At the
linear geometry, the RHF ground state is of the form
$(1a_{1})^{2}\,(2a_{1})^{2}\,(1b_{2})^{2}$. While this state has A1 symmetry,
we refer to it as the “HF b2” state to denote the symmetry of the highest
occupied molecular orbital. At $x=2$, on the other hand, the RHF ground state
is of the form $(1a_{1})^{2}\,(2a_{1})^{2}\,(3a_{1})^{2}$. This state is also
of A1 symmetry, and we refer to it as the “HF a1” state. These two RHF
solutions cross at $x\approx 1.5$ (see the top panel of Figure 5). Previous
studies[4] showed that complex RHF can smoothly connect the two
configurations, but is far from the exact answer around the crossing region.
Note that complex conjugation symmetry only spontaneously breaks near $x=1.5$.

At the FCI level, meanwhile, there are three low-energy singlets. Two are of
A1 symmetry and reflect the two RHF solutions we have just described, and the
third is of B2 symmetry and is essentially the open-shell singlet arising from
occupied the frontier $B_{2}$ and $A_{1}$ orbitals once each. Where the RHF
states cross near $x=1.5$, the FCI instead has an avoided crossing, and this
avoided crossing implies strong correlation. Complicating things somewhat is
that where the exact ground state is of A1 symmetry across most of the
insertion pathway, the B2 open-shell singlet is the ground state near $x=1.5$.

Figure 5: Total energies in the BeH2 insertion. Top panel: Comparison between
HF and FCI. Middle panel: Comparison between PHF and FCI. Bottom panel:
Energies in the three distinct KRHF solutions.

The middle panel of Figure 5 shows the energy of various PHF states in
comparison to the FCI benchmark. Comparing the top and middle panels reveals
that, though complex conjugation projected RHF (KRHF) is far from exact, it is
still notably more accurate than is RHF. Unfortunately, KRHF has a shoulder at
$x\approx 1.35$ and another at $x\approx 1.65$. This is due to the presence of
the FCI B2 solution. The complex RHF has, as its highest occupied molecular
orbital, a linear combination of a1 and $b_{2}$ character so that complex RHF
has broken point group symmetry. Complex conjugation projection tends also to
restore point group symmetry, and the ground state in the region of the
shoulder is of B2 symmetry, not of A1 symmetry as we see elsewhere. The bottom
panel of Fig. 5 makes this point clearly. We see that there are three distinct
KRHF solutions. For $x\lesssim 1.35$ and $x\gtrsim 1.63$, the KRHF is of A1
symmetry corresponding to the two distinct A1 RHF solutions. For $x$ between
these two values, KRHF instead gives a B2 state. We can follow these three
distinct KRHF solutions beyond the point at which they are the KRHF ground
state, but at some point we are unable to locate them.

The solution to this problem is to include point group symmetry projection
along with complex conjugation projection. These give us the curves marked in
the middle panel of Fig. 5 as A1 KRHF and B2 KRHF, where we have projected
complex conjugation and point group symmetry together and optimized the
reference determinant accordingly. Including point group projection yields
well-behaved curves which are nearly parallel to the correspinding FCI
references.

Figure 6 shows the total energies (top panel) and errors with respect to FCI
(bottom panel) for various coupled cluster methods. The restricted CCSD
(RCCSD) data are obtained by doing (real) RCCSD based on the two A1 RHF
determinants and taking the lower of the two energies at each point. Since
this RCCSD has A1 symmetry everywhere, we compare it to the A1 FCI, and we see
that it is fairly accurate except in the immediate vicinity of $x=1.5$ where
the correlations are strongest. The complex-conjugation projected RCCSD
(KRCCSD) curve is likewise obtained by doing KRCCSD curves starting from the
distinct KRHF solutions and taking the lowest. Like RCCSD, it is highly
accurate, but because the KRHF reference is of B2 symmetry for intermediate
values of $x$, the KRCCSD curve approximates the FCI B2 state in this region.
Including point group and complex conjugation projection to get A1 KRCCSD or
B2 KRCCSD gives results which agree with the corresponding FCI with errors
less than 2 mEh.

Figure 6: Coupled cluster energies in the BeH2 insertion. Top panel: Total
energies, compared to FCI. Bottom panel: Energy error of CCSD and projected
CCSD compared to FCI.

##  V Discussion and Conclusion

###  V.1 Discussion

This work shows a systematic way to restore complex conjugation symmetry at
both the Hartree-Fock and coupled cluster levels. It is easy to check that
when the symmetry is not broken, these theories go back to traditional HF and
CC. The disentangled cluster formalism and power series approximation for the
discrete symmetry yield satisfactory results even though the symmetries
involved are discrete instead of continuous.

On the other hand, as discussed elsewhere,[12, 13] projected CC has a problem
with what we call gauge invariances. These are numerical invariances of the
algebraic equations that leave observables unchanged. Although gauge
invariances do not affect the final result, they still make the PCCSD result
not unique when using approximation methods. In this work, we choose the
projection operator to be $1+K$ to fix the gauge in complex conjugation, but
whether it is the best choice still needs to be determined. When introducing
spin-flip symmetry, we have an extra spin gauge mode, and this mode causes
trouble in the convergence of the PCC equations. The spin-flip projection only
contains two operators, but we need to use large damping factors to slow down
changes in CC amplitudes. Otherwise, it oscillates dramatically.

###  V.2 Conclusion

Complex wave functions are useful tools in describing degenerate systems and
contain richer information than do real wave functions. With proper
projection, we can eliminate the unphysical part of complex wave functions and
achieve superior results. Coupled cluster is a sound wave function that
captures dynamic correlation, so combining complex conjugation projection and
coupled cluster is natural. Benefiting from the projected coupled cluster
theory, we can build an effective method to implement complex conjugation
projected CCSD. Also, we add spin flip projection under the same framework,
which enables us to restore time-reversal symmetry as well. These projected
wave functions successfully solve some well-known multi-reference problems and
achieve results of roughly chemical accuracy without requiring a description
of higher particle-hole excitations beyond doubles.

##  VI Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

###### Acknowledgements.

This work was supported by the U.S. National Science Foundation under Grant
No. CHE-1762320. G.E.S. is a Welch Foundation Chair (Grant No. C-0036).

##  Appendix A Gradient for projected HF

In this appendix, we provide a detailed discussion of the optimization of
symmetry projected HF states based on a Thouless parametrization. In Eqn 11,
we treat $z$ and $z^{*}$ as independent variables, the gradient is

$G_{ai}=\frac{\partial E}{\partial
z^{*}_{ai}}=\frac{\langle\Phi|c_{i}^{\dagger}c_{a}\,e^{Z^{\dagger}}\,\overleftarrow{P_{K}}\,P^{\dagger}_{F}\,(H-E)\,P_{F}\,P_{K}\,e^{Z}|\Phi\rangle}{\langle\Phi|e^{Z^{\dagger}}\,\overleftarrow{P_{K}}\,P^{\dagger}_{F}\,P_{F}\,P_{K}\,e^{Z}|\Phi\rangle}$
(23)

Since we choose the reference state to be real, complex conjugation operator
will only change the parameter $z,z^{*}$ and leave the reference unchanged.

The gradient is made of two parts, the kernel for the overlap-like part is

$\frac{\langle\Phi|c_{i}^{\dagger}c_{a}\,e^{Z^{\dagger}}\,R\,e^{Z}|\Phi\rangle}{\langle\Phi|e^{Z^{\dagger}}\,R\,e^{Z}|\Phi\rangle}=\rho_{ai}$
(24)

and for the Hamiltonian-like part is

$\frac{\langle\Phi|c_{i}^{\dagger}c_{a}\,e^{Z^{\dagger}}\,H\,R\,e^{Z}|\Phi\rangle}{\langle\Phi|e^{Z^{\dagger}}\,R\,e^{Z}|\Phi\rangle}=h\,\rho_{ai}+[(1-\rho)\,(h+\Gamma)\,\rho]_{ai}$
(25)

Here $R$ is the operator in the projection operators, and $h$ and $\Gamma$ are
defined as following

$\displaystyle
h=\frac{\langle\Phi|e^{Z^{\dagger}}\,H\,R\,e^{Z}|\Phi\rangle}{\langle\Phi|e^{Z^{\dagger}}\,R\,e^{Z}|\Phi\rangle}$
(26) $\displaystyle\Gamma_{ik}=\sum_{jl}V_{ijkl}\,\rho_{lj}$ (27)

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# INDEX, PRIME IDEAL FACTORIZATION IN SIMPLEST QUARTIC FIELDS and counting
their discriminants.

Mohammed Seddik  Mohammed Seddik
Université d’Évry Val d’Essonne
Laboratoire de Mathmatiques et Modélisation d’Évry (UMR 8071)
I.B.G.B.I., 23 Bd. de France, 91037 Évry Cedex, France
<EMAIL_ADDRESS>
###### Abstract.

We consider the simplest quartic number fields $\mathbb{K}_{m}$ defined by the
irreducible quartic polynomials

$x^{4}-mx^{3}-6x^{2}+mx+1,$

where $m$ runs over the positive rational integers such that the odd part of
$m^{2}+16$ is squarefree. In this paper, we study the common index divisor
$I(\mathbb{K}_{m})$ and determine explicitly the prime ideal decomposition for
any prime number in any simplest quartic number fields $\mathbb{K}_{m}$. On
the other hand, we establish an asymptotic formula for the number of simplest
quartic fields with discriminant $\leq x$ and given index.

###### Key words and phrases:

Common factor of indices, common divisor of values of polynomials, prime ideal
factorization, cyclic quartic fields, discriminant.

###### 2000 Mathematics Subject Classification:

Primary : 11R04,11R33, 13F20

∗ Corresponding Author: Mohammed Seddik<EMAIL_ADDRESS>
##  1\. Introduction and preliminaries

Let $\mathbb{K}$ be a number field. We start with given most important known
results on the common factor of indices $I(\mathbb{K})$, the prime ideal
factorization and the asymptotic formula for number fields with discriminant
$\leq x$.

###  1.1. Index $I(\mathbb{K})$ for number fields $\mathbb{K}$

Let $\mathbb{K}$ be a number field of degree $n$ over $\mathbb{Q}$ and let
$\mathbb{O_{K}}$ be its ring of integers. Denote by
$\widehat{\mathbb{O}}_{\mathbb{K}}$ the set of primitive elements of
$\mathbb{O_{K}}$. For any $\theta\in\mathbb{O_{K}}$ we denote $F_{\theta}(x)$
the characteristic polynomial of $\theta$ over $\mathbb{Q}$. Let
$D_{\mathbb{K}}$ be the discriminant of $\mathbb{K}$. It is well known that if
$\theta\in\widehat{\mathbb{O}}_{\mathbb{K}}$, the discriminant of
$F_{\theta}(x)$ has the form

(1.1) $D(\theta)=I(\theta)^{2}D_{\mathbb{K}},$

where $I(\theta)=(\mathbb{O_{K}}:\mathbb{Z}[\theta])$ is called the index of
$\theta$. Let

(1.2)
$I(\mathbb{K})=\underset{\theta\in\widehat{\mathbb{O}}_{\mathbb{K}}}{\text{gcd}}\;I(\theta).$

A prime number $p$ is called a common factor of indices if $p\mid
I(\mathbb{K})$.

The knowledge of the index $I(\mathbb{K})$ makes it possible to find the
explicit decomposition of the prime numbers in the fields $\mathbb{K}$ : If
the prime number $p\nmid I(\mathbb{K})$, by equation (1.2) there exist a
primitive integer $\theta$ where $p\nmid I(\theta)$ and by Dedekind’s theorem
[5, §18] we explicitly have the factorization of $p$ using $\theta$. However,
if $p$ is a common factor divisor, the prime ideal decomposition in
$\mathbb{O_{K}}$ is more difficult.

Let us recall the statement of Dedekind’s theorem. Let
$\mathbb{K}=\mathbb{Q}(\theta)$ be an algebraic number field with
$\theta\in\mathbb{O_{\mathbb{K}}}$. Let $p$ be a rational prime. Let

$f(x)=\textrm{Irr}_{\mathbb{Q}}(x,\theta)\in\mathbb{Z}[x].$

We consider the canonical surjection map
$\mathbb{Z}[x]\to\mathbb{Z}/p\mathbb{Z}[x]$. We write

$\bar{f}(x)=g_{1}(x)^{e_{1}}\cdots g_{r}(x)^{e_{r}},$

where $g_{1}(x),\cdots,g_{r}(x)$ are distinct monic irreducible polynomials in
$\mathbb{Z}/p\mathbb{Z}[x]$ and $e_{1},\cdots,e_{r}$ are positive integers.
For $i=1,2,\ldots,r$ denote by $f_{i}(x)$ any monic polynomial of
$\mathbb{Z}[x]$ such that $\bar{f}_{i}=g_{i}$. We then set

$P_{i}=<p,f_{i}(\theta)>.$

If $I(\theta)\not\equiv 0\mod p$ then we have $P_{1},\cdots,P_{r}$ are
distinct prime ideals of $\mathbb{O_{\mathbb{K}}}$ with

$p\mathbb{O}_{\mathbb{K}}={P_{1}^{e_{1}}}\cdots{P_{r}^{e_{r}}},$

and

$N(P_{i})=p^{\deg(f_{i})}.$

In the following we review some known results on the computation of the index
$I(\mathbb{K})$.

$\bullet$ If $\mathbb{K}$ is a quadratic field $(n=2)$, by classical number
theory on quadratic fields, one can show that $I(\mathbb{K})=1$.
$\bullet$ In case $\mathbb{K}$ is a cubic field $(n=3)$ , Engstrom [6] showed
that $I(\mathbb{K})=1$ or $2$. Llorente and Nart in [14, Theorem 1] determine
the type of decomposition of the rational primes and in [14, Theorem 4] give a
necessary and sufficient condition for the index of $\mathbb{K}$ to be $2.$
Moreover, in the paper [19] Spearman and Williams give the explicit prime
ideal factorization of $2$ in cubic fields with index $2$.
$\bullet$ In the case $\mathbb{K}$ is a quartic field $(n=4)$, Engstrom [6]
showed that

(1.3) $I(\mathbb{K})=1,2,3,4,6\;\text{or}\;12.$

$\bullet$ If the field $\mathbb{K}$ is a cyclic quartic field , Spearman and
Williams [20] showed that $I(\mathbb{K})$ assumes all of these values and they
give necessary and sufficient conditions for each to occur and find an
asymptotic formula for the number of cyclic quartic fields with discriminant
$\leq x$ and $I(\mathbb{K})=i$ for each $i\in\\{1,2,3,4,6,12\\}$,

$N(x;i)=\alpha_{i}x^{\frac{1}{2}}+O(x^{\frac{1}{3}}log^{3}x),$

where

$\displaystyle\alpha_{1}$ $\displaystyle\approx
0.0970153,\quad\alpha_{2}\approx 0.0067627,\quad\alpha_{3}\approx 0.0101764,$
$\displaystyle\alpha_{4}$ $\displaystyle\approx
0.0067627,\quad\alpha_{6}\approx 0.0006321,\quad\alpha_{12}\approx 0.0006321.$

$\bullet$ Funakura [7, Theorem 5] showed that in the case of a pure quartic
field, we have $I(\mathbb{K})=1,2$.
$\bullet$ In the pure quartic field , Spearman and Williams [21] gives the
explicit prime ideal factorization of $2$ when the index is equal to $2$ .

###  1.2. Counting discriminants with given index

The general problem in algebraic number theory is that of counting the number
of fields by discriminant. If we let $N_{n}(x)$ denote the number of fields of
degree $n$ over $\mathbb{Q}$ whose discriminants do not exceed $x$ in absolute
value. Then there is a conjecture that

$\lim_{x\to\infty}\frac{N_{n}(x)}{x}$

exist and is non-zero. This conjecture is proved in many special cases:

$\bullet$ For quadratic field, the problem is simple we have

$N_{2}(x)=\sum_{|d|\leq x\atop d\equiv
1\;mod\;4}1+\sum_{|d|\leq\frac{x}{4}\atop d\equiv
2,3\;mod\;4}1\sim\frac{6}{\pi^{2}}x.$

$\bullet$ For cyclic cubic fields H. Cohn [4] and Cohen-Diaz y Diaz in [3,
p.577, §3] showed that

$N_{3}(x)\sim C_{3}x^{1/2}$

where

$C_{3}=\frac{11\sqrt{3}}{36\pi}\prod_{p\equiv 1\;mod\;6\atop p\textrm{ prime
}}\left(1-\frac{2}{p(p+1)}\right).$

$\bullet$ For cyclic quartic fields Baily [2, p. 209, Theorem 9] and its
revised form by many authors see [13], [18] and [3, p.580, §5] by differents
methods, as follows

$N_{4}(x)\sim C_{4}x^{1/2}$

where

$C_{4}=\frac{3}{\pi^{2}}\left((1+\frac{\sqrt{2}}{24})\prod_{p\equiv
1\;mod\;4\atop p\textrm{ prime
}}\left(1+\frac{2}{p^{3/2}+p^{1/2}}\right)-1\right).$

###  1.3. Summary of the paper

In this paper, we investigate the family of the simplest quartic fields which
are defined by adjunction to $\mathbb{Q}$ of a root of the polynomial

$P_{m}=x^{4}-mx^{3}-6x^{2}+mx+1,\quad m\in\mathbb{Z}^{+}$

where $m^{2}+16$ is not divisible by an odd square. It is easy to see that $m$
may be specified greater than zero as $P_{m}$ and $P_{-m}$ generate the same
extension.

M.N. Gras in [9] proved that those polynomials are reducibles precisely when
$m^{2}+16$ is a square, which occurs only for excluded cases $m=0,\,3$, and
show that the form $m^{2}+16$ represents infinitely many square-free integers.
Olajos [17] proved that $\mathbb{K}_{m}$ admits power integral bases only for
$m=2,\,4$ and he gave all generators of power integral bases. Recently, Gaál
and Pentráyani [8] compute the minimal index of the simplest quartic fields.

The purpose of this paper :
$\bullet$ We study the common indices $I(\mathbb{K}_{m})$ and the prime ideal
decomposition is determined explicitly,
$\bullet$ We establish an asymptotic formula for the number of the simplest
quartic fields with given index and the discriminants less than $x$.

##  2\. Statement of main results

We state our main results.

###  2.1. Computation of the index $I(\mathbb{K}_{m})$ and prime decomposition

######  Theorem 2.1.

Let $m\in\mathbb{Z}^{+},\;m\neq 0,3$ and $\mathbb{K}_{m}=\mathbb{Q}(\theta)$
where $\theta$ be a root of $P_{m}$. Then we have

  1. (1)

$I(\mathbb{K}_{m})=\begin{cases}2&\text{if \; $m$ odd},\\\ 1&\text{if \; $m$
even}.\end{cases}$

  2. (2)

The prime ideal factorization of $2$ in $\mathbb{K}_{m}$ with index $2$ is

$2\mathbb{O_{K}}_{m}=<2,\frac{1+\sqrt{m^{2}+16}}{2}><2,\frac{1-\sqrt{m^{2}+16}}{2}>.$

  3. (3)

The prime ideal factorization of $2$ in $\mathbb{K}_{m}$ with index $1$ is as
follows:

    1. 1)

If $v_{2}(m)=1$, then,

$2\mathbb{O_{K}}_{m}=<2,\frac{m\theta^{3}+10\theta^{2}-m\theta+6}{4}>^{2}.$

    2. 2)

If $v_{2}(m)=2$, then,

$2\mathbb{O_{K}}_{m}=<2,\frac{1+\theta+\theta^{2}+\theta^{3}}{4}>^{4}.$

    3. 3)

If $v_{2}(m)=3$, then,

$2\mathbb{O_{K}}_{m}=<2,\frac{(m^{3}+25m+4)\theta^{3}+(5m^{2}+168)\theta^{2}-(m^{3}+21m-56)\theta-m^{2}+8}{16}>^{2}.$

    4. 4)

If $v_{2}(m)\geq 4$, then,

$2\mathbb{O_{K}}_{m}=<2,\frac{2+7\theta+\theta^{3}}{4}>^{2}<2,\frac{5+7\theta+\theta^{3}}{4}>^{2}.$

######  Proposition 2.1.

Let $m\in\mathbb{Z}^{+},\;m\neq 0,3$ and $\mathbb{K}_{m}=\mathbb{Q}(\theta)$
where $\theta$ be a root of $P_{m}$. Then

$I(\theta)=\begin{cases}2&\text{if \; $v_{2}(m)=0$},\\\ 2^{2}&\text{if \;
$v_{2}(m)=1$ },\\\ 2^{3}&\text{if \; $v_{2}(m)=2$},\\\ 2^{4}&\text{if \;
$v_{2}(m)\geq 3$}.\\\ \end{cases}$

######  Remark.

Note that we can show that for any $p$ odd prime, we acn give explicitly the
factorization of $p$ using $\theta$. To do it, we need to factorize
$P_{m}\;mod\;p$ and we use Dedekind’s theorem.

###  2.2. Asymptotic number of simplest quartic fields with discriminant $\leq
x$ and given index

Let $\mathbb{K}_{m}=\mathbb{Q}(\theta)$ where $\theta$ be a root of $P_{m}$
(simplest quartic fields).
We define for a positive integer $i$

(2.1) $N(x,i)=\text{number of $\mathbb{K}_{m}$ with $D(\mathbb{K}_{m})\leq x$
and $I(\mathbb{K}_{m})=i$}.$

We state our second main result.

######  Theorem 2.2.

For $i=1,2,3,4,6,12$ we have

  1. (1)

$N(x,i)=0$ for $i\neq 1,2$.

  2. (2)

For $i=1,2$, we have the asymptotic formulas

$N(x,i)\sim C_{i}x^{\frac{1}{6}},$

where

$C_{1}=\left(\frac{1}{4\sqrt[3]{4}}+\frac{1}{4\sqrt[3]{2}}+\frac{1}{4}\right)\prod_{p\equiv
1\;mod\;4\atop p\textrm{ prime }}\left(1-\frac{2}{p^{2}}\right)$

and

$C_{2}=\frac{1}{4}\prod_{p\equiv 1\;mod\;4\atop p\textrm{ prime
}}\left(1-\frac{2}{p^{2}}\right)$

As corollary, from Theorem 2.2, the number $N(x)$ of simplest quartic fields
$\mathbb{K}_{m}$ with discriminant $D(\mathbb{K}_{m})\leq x$ is given by

$N(x)=N(x,1)+N(x,2)\sim\left(\frac{1}{4\sqrt[3]{4}}+\frac{1}{4\sqrt[3]{2}}+\frac{1}{2}\right)\prod_{p\equiv
1\;mod\;4\atop p\textrm{ prime }}\left(1-\frac{2}{p^{2}}\right)\
x^{\frac{1}{6}},\textrm{ as }x\to\infty$

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