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<div class="section" id="dcft-density-cumulant-functional-theory">
<span id="sec-dcft"></span><span id="index-0"></span><h1>DCFT: Density Cumulant Functional Theory<a class="headerlink" href="#dcft-density-cumulant-functional-theory" title="Permalink to this headline">¶</a></h1>
<p><em>Code author: Alexander Yu. Sokolov, Andrew C. Simmonett, and Xiao Wang</em></p>
<p><em>Section author: Alexander Yu. Sokolov</em></p>
<p><em>Module:</em> <a class="reference internal" href="autodir_options_c/module__dcft.html#apdx-dcft"><span>Keywords</span></a>, <a class="reference internal" href="autodir_psivariables/module__dcft.html#apdx-dcft-psivar"><span>PSI Variables</span></a>, <a class="reference external" href="https://github.com/psi4/psi4public/blob/master/src/bin/dcft">DCFT</a></p>
<div class="section" id="theory">
<span id="sec-dcfttheory"></span><h2>Theory<a class="headerlink" href="#theory" title="Permalink to this headline">¶</a></h2>
<p>Density cumulant functional theory (DCFT) is a density-based <em>ab initio</em> theory
that can compute electronic energies without the use of a wavefunction. The
theory starts by writing the exact energy expression in terms of the one- and
two-particle density matrices (<img class="math" src="_images/math/2bbde098ee69a40395e5320eed2e4b9c1fa7c27b.png" alt="\mbox{\boldmath $\gamma_1$}" style="vertical-align: -4px"/> and <img class="math" src="_images/math/1f22b5e8c05e4615b20bd072bbf18d699874f3b8.png" alt="\mbox{\boldmath $\gamma_2$}" style="vertical-align: -4px"/>):</p>
<div class="math">
<p><img src="_images/math/3c6250bd39cbe3000495eb1dd1c571c5145f649f.png" alt="E &= h_p^q \gamma_q^p + \frac{1}{2} g_{pq}^{rs} \gamma_{rs}^{pq}"/></p>
</div><p>Here we used Einstein convention for the summation over the repeated indices,
<img class="math" src="_images/math/e0807bb5ec79844f887446c11830ab6eb0fc9ac7.png" alt="h_p^q" style="vertical-align: -8px"/> and <img class="math" src="_images/math/f39bced979a7abed471b4c71314732815d8b172b.png" alt="g_{pq}^{rs}" style="vertical-align: -8px"/> are the standard one- and two-electron integrals,
<img class="math" src="_images/math/5ca5145abaf15eb0b4abc24eeadd3b24e9e43fe3.png" alt="\gamma_p^q" style="vertical-align: -8px"/> and <img class="math" src="_images/math/92a3fb148e9622b891d4573ca3ab34441b8585be.png" alt="\gamma_{pq}^{rs}" style="vertical-align: -8px"/> are the elements of <img class="math" src="_images/math/2bbde098ee69a40395e5320eed2e4b9c1fa7c27b.png" alt="\mbox{\boldmath $\gamma_1$}" style="vertical-align: -4px"/> and <img class="math" src="_images/math/1f22b5e8c05e4615b20bd072bbf18d699874f3b8.png" alt="\mbox{\boldmath $\gamma_2$}" style="vertical-align: -4px"/>,
respectively. Naively, one might expect that it is possible to minimize the
energy functional in the equation above and obtain the exact energy. This is,
however, not trivial, as the density matrix elements <img class="math" src="_images/math/5ca5145abaf15eb0b4abc24eeadd3b24e9e43fe3.png" alt="\gamma_p^q" style="vertical-align: -8px"/> and
<img class="math" src="_images/math/92a3fb148e9622b891d4573ca3ab34441b8585be.png" alt="\gamma_{pq}^{rs}" style="vertical-align: -8px"/> cannot be varied arbitrarily, but must satisfy some
conditions that make sure that the density matrices are N-representable, <em>i.e.</em>
correspond to an antisymmetric N-electron wavefunction. Unfortunately, no
simple set of necessary and sufficient N-representability conditions are known,
and some of the known conditions are not easily imposed. In addition, the lack
of separability of the density matrices may result in the loss of
size-consistency and size-extensivity. In DCFT, one takes a different route and
replaces <img class="math" src="_images/math/1f22b5e8c05e4615b20bd072bbf18d699874f3b8.png" alt="\mbox{\boldmath $\gamma_2$}" style="vertical-align: -4px"/> in favor of its two-particle density cumulant:</p>
<div class="math">
<p><img src="_images/math/af48f2abd83687ed2beedb66ddc244ab873bd30c.png" alt="\lambda_{pq}^{rs} = \gamma_{pq}^{rs} - \gamma_p^r \gamma_q^s + \gamma_p^s \gamma_q^r"/></p>
</div><p>The one-particle density matrix is separated into its idempotent part
<img class="math" src="_images/math/f1a995a68df833cb4486506097e7e3355943c8c1.png" alt="\mbox{\boldmath $\kappa$}" style="vertical-align: 0px"/> and a correction <img class="math" src="_images/math/34e0b75ab3a1eaf13df0a6ad5911b3baf93991aa.png" alt="\mbox{\boldmath $\tau$}" style="vertical-align: 0px"/>:</p>
<div class="math">
<p><img src="_images/math/e9934931d87bfa83317f0ba301cfb9c7038ffbc6.png" alt="\gamma_p^q = \kappa_p^q + \tau_p^q"/></p>
</div><p>The idempotent part of <img class="math" src="_images/math/2bbde098ee69a40395e5320eed2e4b9c1fa7c27b.png" alt="\mbox{\boldmath $\gamma_1$}" style="vertical-align: -4px"/> corresponds to a mean-field Hartree-Fock-like density,
while the non-idempotent correction <img class="math" src="_images/math/34e0b75ab3a1eaf13df0a6ad5911b3baf93991aa.png" alt="\mbox{\boldmath $\tau$}" style="vertical-align: 0px"/>
depends on the density cumulant and describes the electron correlation effects.
Inserting the above two equations into the energy expression, we obtain:</p>
<div class="math">
<p><img src="_images/math/6e071fdcc2774aafe4fc3c87143c86253817d7c3.png" alt="E_{DCFT} &= \frac{1}{2} \left( h_p^q + f_p^q \right) \gamma_q^p + \frac{1}{4} \bar{g}_{pq}^{rs} \lambda_{rs}^{pq}"/></p>
</div><p>where the antisymmetrized two-electron integrals and the generalized Fock operator
matrix elements were defined as follows:</p>
<div class="math">
<p><img src="_images/math/6d9213d38079872acdc912dbbd05a8b599563a99.png" alt="\bar{g}_{pq}^{rs} = g_{pq}^{rs} - g_{pq}^{sr}"/></p>
</div><div class="math">
<p><img src="_images/math/80e0d70961188e683c80fef17120b3a1dbaab0b5.png" alt="f_p^q = h_p^q + \bar{g}_{pr}^{qs} \gamma_{s}^{r}"/></p>
</div><p>Energy functional <img class="math" src="_images/math/08612bfe17abc6a0273a9d61f3c874813b162a95.png" alt="E_{DCFT}" style="vertical-align: -3px"/> has several important properties. First,
the energy is now a function of two sets of independent parameters, the
idempotent part of <img class="math" src="_images/math/2bbde098ee69a40395e5320eed2e4b9c1fa7c27b.png" alt="\mbox{\boldmath $\gamma_1$}" style="vertical-align: -4px"/> (<img class="math" src="_images/math/f1a995a68df833cb4486506097e7e3355943c8c1.png" alt="\mbox{\boldmath $\kappa$}" style="vertical-align: 0px"/>) and the density cumulant
(<img class="math" src="_images/math/74f3f1715099b8e0a558272aeb68953fe74b1bed.png" alt="\mbox{\boldmath $\lambda_2$}" style="vertical-align: -3px"/>). As a result, the energy functional is Hermitian,
which is important for the evaluation of the molecular properties. The additive
separability of the density cumulant guarantees that all of the DCFT methods
are size-extensive and size-consistent. Furthermore, the N-representability
problem is now greatly simplified, because the idempotent part of <img class="math" src="_images/math/2bbde098ee69a40395e5320eed2e4b9c1fa7c27b.png" alt="\mbox{\boldmath $\gamma_1$}" style="vertical-align: -4px"/> is
N-representable by construction. One only needs to worry about the
N-representability of the density cumulant, which is a relatively small part of
<img class="math" src="_images/math/1f22b5e8c05e4615b20bd072bbf18d699874f3b8.png" alt="\mbox{\boldmath $\gamma_2$}" style="vertical-align: -4px"/>.</p>
<p>In order to obtain the DCFT energy, two conditions must be satisfied:</p>
<p>1) The energy must be stationary with respect to a set of orbitals. This can be done by
diagonalizing the generalized Fock operator (as in the DC-06 and DC-12 methods, see below),
which introduces partial orbital relaxation, or by fully relaxing the orbitals and minimizing the entire energy expression
(as in the ODC-06 and ODC-12 methods).</p>
<p>2) The energy must be stationary with respect to the variation of the density
cumulant <img class="math" src="_images/math/74f3f1715099b8e0a558272aeb68953fe74b1bed.png" alt="\mbox{\boldmath $\lambda_2$}" style="vertical-align: -3px"/>, constrained to N-representability conditions.</p>
<p>Making the energy stationary requires solution of two sets of coupled
equations for orbitals and density cumulant, respectively (also known as
residual equations). At the present moment, three different algorithms for the
solution of the system of coupled equations are available (see section
<a class="reference internal" href="#sec-dcftalgorithms"><span>Iterative Algorithms</span></a> for details).</p>
<p>Publications resulting from the use of the DCFT code should cite contributions
listed <a class="reference internal" href="introduction.html#intro-dcftcitations"><span>here</span></a>.</p>
</div>
<div class="section" id="methods">
<span id="sec-dcftmethods"></span><h2>Methods<a class="headerlink" href="#methods" title="Permalink to this headline">¶</a></h2>
<p>Currently five DCFT methods (functionals) are available: DC-06, DC-12, ODC-06, ODC-12, and ODC-13. The first four
methods use approximate N-representability conditions derived from
second-order perturbation theory and differ in the description of the
correlated (non-idempotent) part <img class="math" src="_images/math/34e0b75ab3a1eaf13df0a6ad5911b3baf93991aa.png" alt="\mbox{\boldmath $\tau$}" style="vertical-align: 0px"/> of the one-particle density
matrix and orbital optimization. While in the DC-06 and ODC-06 methods <img class="math" src="_images/math/34e0b75ab3a1eaf13df0a6ad5911b3baf93991aa.png" alt="\mbox{\boldmath $\tau$}" style="vertical-align: 0px"/> is derived from the density cumulant
in an approximate way (labelled by ‘06’), the DC-12 and ODC-12 methods derive this contribution exactly, and
take full advantage of the N-representability conditions (which is denoted by ‘12’). The corresponding DC and ODC methods
have similar description of the <img class="math" src="_images/math/2bbde098ee69a40395e5320eed2e4b9c1fa7c27b.png" alt="\mbox{\boldmath $\gamma_1$}" style="vertical-align: -4px"/> N-representability, but differ in describing the orbital relaxation:
the former methods account for the relaxation only partially, while the latter fully relax the orbitals.
Both DC-06 and DC-12 methods have similar computational cost, same is true when comparing ODC-06 and ODC-12.
Meanwhile, the DC methods are generally more efficient than their ODC analogs, due to a more expensive orbital update step
needed for the full orbital optimization. In the ODC-13 method, the third- and
fourth-order N-representability conditions are used for the density cumulant
and the correlated contribution <img class="math" src="_images/math/34e0b75ab3a1eaf13df0a6ad5911b3baf93991aa.png" alt="\mbox{\boldmath $\tau$}" style="vertical-align: 0px"/>, respectively,
and the orbitals are variationally optimized.
For most of the applications, it is recommended to use the ODC-12 method, which
provides an optimal balance between accuracy and efficiency, especially for
molecules with open-shell character. If highly accurate results are desired, a
combination of the ODC-13 method with a three-particle energy correction
[<img class="math" src="_images/math/a90dc71e91465be5684e040e53e54d13841a1673.png" alt="\mbox{ODC-13$(\lambda_3)$}" style="vertical-align: -4px"/>] can be used (see below).
For the detailed comparison of the quality of these methods we refer
users to our <a class="reference internal" href="introduction.html#intro-dcftcitations"><span>publications</span></a>.</p>
<p>The DCFT functional can be specified by the <a class="reference internal" href="autodoc_glossary_options_c.html#term-dcft-functional-dcft"><span class="xref std std-term">DCFT_FUNCTIONAL</span></a> option. The
default choice is the ODC-12 functional. In addition to five methods listed
above, <a class="reference internal" href="autodoc_glossary_options_c.html#term-dcft-functional-dcft"><span class="xref std std-term">DCFT_FUNCTIONAL</span></a> option can be set to CEPA0 (coupled electron
pair approximation zero, equivalent to linearized coupled cluster doubles
method, LCCD). CEPA0 can be considered as a particular case of the DC-06 and DC-12
methods in the limit of zero non-idempotency of <img class="math" src="_images/math/2bbde098ee69a40395e5320eed2e4b9c1fa7c27b.png" alt="\mbox{\boldmath $\gamma_1$}" style="vertical-align: -4px"/>. This option has a limited
functionality and should only be used for test purposes. For the production-level CEPA0 code, see the
description of the OCC section of the manual. The DCFT code can also be used to
compute the <img class="math" src="_images/math/543b8acaf2b58ab61f743cc07b659e2e97246620.png" alt="(\lambda_3)" style="vertical-align: -4px"/> energy correction that perturbatively accounts
for three-particle correlation effects, similarly to the (T) correction in coupled
cluster theory. Computation of the <img class="math" src="_images/math/543b8acaf2b58ab61f743cc07b659e2e97246620.png" alt="(\lambda_3)" style="vertical-align: -4px"/> correction can be
requested by setting the <a class="reference internal" href="autodoc_glossary_options_c.html#term-three-particle-dcft"><span class="xref std std-term">THREE_PARTICLE</span></a> option to PERTURBATIVE. A
combination of the ODC-13 functional with the <img class="math" src="_images/math/543b8acaf2b58ab61f743cc07b659e2e97246620.png" alt="(\lambda_3)" style="vertical-align: -4px"/> correction
[denoted as <img class="math" src="_images/math/a90dc71e91465be5684e040e53e54d13841a1673.png" alt="\mbox{ODC-13$(\lambda_3)$}" style="vertical-align: -4px"/>] has been shown to provide highly
accurate results for open-shell molecules near equilibrium geometries.</p>
<p>At the present moment, all of the DCFT methods support unrestricted reference
orbitals (<a class="reference internal" href="autodoc_glossary_options_c.html#term-reference-scf"><span class="xref std std-term">REFERENCE</span></a> = UHF), which can be used to perform energy and
gradient computations for both closed- and open-shell molecules. In addition,
the ODC-06 and ODC-12 methods support restricted reference orbitals
(<a class="reference internal" href="autodoc_glossary_options_c.html#term-reference-scf"><span class="xref std std-term">REFERENCE</span></a> = RHF) for the energy and gradient computations of
closed-shell molecules. Note that in this case restricted reference orbitals
are only available for <a class="reference internal" href="autodoc_glossary_options_c.html#term-algorithm-dcft"><span class="xref std std-term">ALGORITHM</span></a> = SIMULTANEOUS.</p>
</div>
<div class="section" id="iterative-algorithms">
<span id="sec-dcftalgorithms"></span><h2>Iterative Algorithms<a class="headerlink" href="#iterative-algorithms" title="Permalink to this headline">¶</a></h2>
<p>As explained in the <a class="reference internal" href="#sec-dcfttheory"><span>Theory</span></a> section, in order to obtain the DCFT energy one
needs to solve a system of coupled equations for orbitals and density
cumulant. At the present moment three iterative algorithms for the solution of the
equations are available. The choice of the algorithm is controlled using the
<a class="reference internal" href="autodoc_glossary_options_c.html#term-algorithm-dcft"><span class="xref std std-term">ALGORITHM</span></a> option.</p>
<p>1) Simultaneous algorithm (<a class="reference internal" href="autodoc_glossary_options_c.html#term-algorithm-dcft"><span class="xref std std-term">ALGORITHM</span></a> = SIMULTANEOUS, currently the default).
In this algorithm the DCFT equations are solved in macroiterations.
Each macroiteration consists of a single iteration of the cumulant update
followed by a single iteration of the orbital update and orbital transformation
of the integrals. The macroiterations are repeated until the simultaneous
convergence of the cumulant and orbitals is achieved.
Convergence of the simultaneous algorithm is accelerated using the
DIIS extrapolation technique.</p>
<p>2) Two-step algorithm (can be invoked by setting the <a class="reference internal" href="autodoc_glossary_options_c.html#term-algorithm-dcft"><span class="xref std std-term">ALGORITHM</span></a> option to
TWOSTEP). In the two-step algorithm each macroiteration consists of two sets of
microiterations. In the first set, the density cumulant equations are solved
iteratively, while the orbitals are kept fixed. After the density cumulant is
converged, the second set of microiterations is performed for the
self-consistent update of the orbitals with the fixed density cumulant. Each
macroiteration is completed by performing the orbital transformation of the
integrals. As in the simultaneous algorithm, the DIIS
extrapolation is used to accelerate convergence. Two-step algorithm is
only available for the DC-06 and DC-12 methods.</p>
<p>3) Quadratically-convergent algorithm (set <a class="reference internal" href="autodoc_glossary_options_c.html#term-algorithm-dcft"><span class="xref std std-term">ALGORITHM</span></a> to QC). The
orbital and cumulant update equations are solved using the Newton-Raphson
method. Each macroiteration of the quadratically-convergent algorithm consists
of a single Newton-Raphson update followed by the orbital transformation
of the integrals. The solution of the Newton-Raphson equations is performed
iteratively using the preconditioned conjugate gradients method, where only the
product of the electronic Hessian with the step vector is computed for
efficiency. By default, the electronic Hessian is build for both the cumulant and orbital
updates and both updates are performed simultaneously. Setting the <a class="reference internal" href="autodoc_glossary_options_c.html#term-qc-type-dcft"><span class="xref std std-term">QC_TYPE</span></a>
option to TWOSTEP will perform the Newton-Raphson update only for the orbitals,
while the equations for the cumulant will be solved using a standard Jacobi update.
If requested by the user (set <a class="reference internal" href="autodoc_glossary_options_c.html#term-qc-coupling-dcft"><span class="xref std std-term">QC_COUPLING</span></a> to TRUE), the electronic Hessian can include
matrix elements that couple the orbitals and the density cumulant.
The computation of these coupling elements increases
the cost of the macroiteration, but usually leads to faster convergence and is
recommended for open-shell systems.
It is important to note that the quadratically-convergent algorithm is not yet fully
optimized and often converges slowly when the RMS of the cumulant or
the orbital gradient is below <img class="math" src="_images/math/663fdd1d2c79aac108d5921116f266c9e2739433.png" alt="10^{-7}" style="vertical-align: -1px"/>.</p>
<p>The choice of the iterative algorithm can significantly affect the cost of the
energy computation. While the two-step algorithm requires a small number of
disk-intensive <img class="math" src="_images/math/ba3bb4e90152921f5602f9ee346b254f2d1ccabf.png" alt="{\cal O}(N^5)" style="vertical-align: -4px"/> integral transformations, the simultaneous
algorithm benefits from a smaller number of expensive <img class="math" src="_images/math/8bd69c62787cd427d7f5e087445d195eeb55403f.png" alt="{\cal O}(N^6)" style="vertical-align: -4px"/>
cumulant updates. As a result, for small closed-shell systems the two-step
algorithm is usually preferred, while for larger systems and molecules with
open-shell character it is recommended to use the simultaneous algorithm.
Efficiency of the simultaneous algorithm can be greatly increased by avoiding
the transformation of the four-index virtual two-electron integrals
<img class="math" src="_images/math/8512a3c56e3c4ada611a7778d1959619dbdee3cf.png" alt="(vv|vv)" style="vertical-align: -5px"/> and computing the terms that involve these integrals in the AO
basis. In order to do that one needs to set the <a class="reference internal" href="autodoc_glossary_options_c.html#term-ao-basis-dcft"><span class="xref std std-term">AO_BASIS</span></a> option to
DISK (currently used by default). For more recommendations on the choice of the algorithm see
<a class="reference internal" href="#sec-dcftrecommend"><span>Recommendations</span></a> section.</p>
</div>
<div class="section" id="analytic-gradients">
<span id="sec-dcftgradients"></span><h2>Analytic Gradients<a class="headerlink" href="#analytic-gradients" title="Permalink to this headline">¶</a></h2>
<p>Analytic gradients are available for the DC-06, ODC-06, ODC-12, and ODC-13 methods.
For DC-06, the evaluation of the analytic gradients requires the solution of the
coupled response equations. Two algorithms are available for their iterative
solution: two-step (default) and simultaneous. These algorithms are similar to those
described for the orbital and cumulant updates in the <a class="reference internal" href="#sec-dcftalgorithms"><span>Iterative Algorithms</span></a>
section and usually exhibit similar efficiency. The choice of the algorithm can
be made using the <a class="reference internal" href="autodoc_glossary_options_c.html#term-response-algorithm-dcft"><span class="xref std std-term">RESPONSE_ALGORITHM</span></a> option. For the DC-12 method the
analytic gradients are not yet available, one has to use numerical gradients to
perform the geometry optimizations. For the ODC-06, ODC-12 and ODC-13 methods no response equations
need to be solved, which makes the computation of the analytic gradients very
efficient. Analytic gradients are not available for the three-particle energy
correction <img class="math" src="_images/math/543b8acaf2b58ab61f743cc07b659e2e97246620.png" alt="(\lambda_3)" style="vertical-align: -4px"/>.</p>
</div>
<div class="section" id="methods-summary">
<span id="sec-dcftmethodsummary"></span><h2>Methods Summary<a class="headerlink" href="#methods-summary" title="Permalink to this headline">¶</a></h2>
<p>The table below summarizes current DCFT code features:</p>
<blockquote>
<div><table border="1" class="docutils" id="table-dcft-methods-summary">
<colgroup>
<col width="26%" />
<col width="44%" />
<col width="6%" />
<col width="7%" />
<col width="17%" />
</colgroup>
<thead valign="bottom">
<tr class="row-odd"><th class="head">Method</th>
<th class="head">Available algorithms</th>
<th class="head">Energy</th>
<th class="head">Gradient</th>
<th class="head">Reference</th>
</tr>
</thead>
<tbody valign="top">
<tr class="row-even"><td>ODC-06</td>
<td>SIMULTANEOUS, QC</td>
<td>Y</td>
<td>Y</td>
<td>RHF/UHF</td>
</tr>
<tr class="row-odd"><td>ODC-12</td>
<td>SIMULTANEOUS, QC</td>
<td>Y</td>
<td>Y</td>
<td>RHF/UHF</td>
</tr>
<tr class="row-even"><td>ODC-13</td>
<td>SIMULTANEOUS, QC</td>
<td>Y</td>
<td>Y</td>
<td>UHF</td>
</tr>
<tr class="row-odd"><td><img class="math" src="_images/math/5aef2104b6b65553085783f0c8b635b0ab4bee71.png" alt="\mbox{ODC-12$(\lambda_3)$}" style="vertical-align: -4px"/></td>
<td>SIMULTANEOUS, QC</td>
<td>Y</td>
<td>N</td>
<td>UHF</td>
</tr>
<tr class="row-even"><td><img class="math" src="_images/math/a90dc71e91465be5684e040e53e54d13841a1673.png" alt="\mbox{ODC-13$(\lambda_3)$}" style="vertical-align: -4px"/></td>
<td>SIMULTANEOUS, QC</td>
<td>Y</td>
<td>N</td>
<td>UHF</td>
</tr>
<tr class="row-odd"><td>DC-06</td>
<td>SIMULTANEOUS, QC, TWOSTEP</td>
<td>Y</td>
<td>Y</td>
<td>UHF</td>
</tr>
<tr class="row-even"><td>DC-12</td>
<td>SIMULTANEOUS, QC, TWOSTEP</td>
<td>Y</td>
<td>N</td>
<td>UHF</td>
</tr>
</tbody>
</table>
</div></blockquote>
<p>Note that for ODC-06 and ODC-12 <a class="reference internal" href="autodoc_glossary_options_c.html#term-reference-scf"><span class="xref std std-term">REFERENCE</span></a> = RHF is only available for
<a class="reference internal" href="autodoc_glossary_options_c.html#term-algorithm-dcft"><span class="xref std std-term">ALGORITHM</span></a> = SIMULTANEOUS. To compute <img class="math" src="_images/math/543b8acaf2b58ab61f743cc07b659e2e97246620.png" alt="(\lambda_3)" style="vertical-align: -4px"/> correction,
the <a class="reference internal" href="autodoc_glossary_options_c.html#term-three-particle-dcft"><span class="xref std std-term">THREE_PARTICLE</span></a> option needs to be set to PERTURBATIVE.</p>
</div>
<div class="section" id="minimal-input">
<span id="sec-dcftmininput"></span><h2>Minimal Input<a class="headerlink" href="#minimal-input" title="Permalink to this headline">¶</a></h2>
<p>Minimal input for the DCFT single-point computation looks like this:</p>
<div class="highlight-python"><div class="highlight"><pre>molecule {
H
H 1 1.0
}
set basis cc-pvdz
energy('dcft')
</pre></div>
</div>
<p>The <code class="docutils literal"><span class="pre">energy('dcft')</span></code> call to <a class="reference internal" href="energy.html#driver.energy" title="driver.energy"><code class="xref py py-func docutils literal"><span class="pre">energy()</span></code></a> executes the DCFT
module, which will first call the SCF module and perform the SCF computation
with RHF reference to obtain a guess for the DCFT orbitals. After SCF is
converged, the program will perform the energy computation using the ODC-12
method. By default, simultaneous algorithm will be used for the solution of
the equations. One can also request to perform geometry
optimization following example below:</p>
<div class="highlight-python"><div class="highlight"><pre>molecule {
H
H 1 1.0
}
set basis cc-pvdz
optimize('dcft')
</pre></div>
</div>
<p>The <code class="docutils literal"><span class="pre">optimize('dcft')</span></code> call will first perform all of the procedures
described above to obtain the ODC-12 energy. After that, the ODC-12 analytic
gradients code will be executed and geometry optimization will be performed.</p>
</div>
<div class="section" id="recommendations">
<span id="sec-dcftrecommend"></span><h2>Recommendations<a class="headerlink" href="#recommendations" title="Permalink to this headline">¶</a></h2>
<p>Here is a list of recommendations for the DCFT module:</p>
<ul class="simple">
<li>Generally, the use of the simultaneous algorithm together with the
<a class="reference internal" href="autodoc_glossary_options_c.html#term-ao-basis-dcft"><span class="xref std std-term">AO_BASIS</span></a> = DISK option is recommended (set by default).</li>
<li>In cases when available memory is insufficient, the use of the <a class="reference internal" href="autodoc_glossary_options_c.html#term-ao-basis-dcft"><span class="xref std std-term">AO_BASIS</span></a> = DISK option
is recommended. This will significantly reduce the memory requirements. However, when
used together with the two-step algorithm, this option can significantly
increase the cost of the energy computation.</li>
<li>In cases when the oscillatory convergence is observed before the DIIS
extrapolation is initialized, it is recommended to increase the threshold for
the RMS of the density cumulant or orbital update residual, below which the
DIIS extrapolation starts. This can be done by setting the
<a class="reference internal" href="autodoc_glossary_options_c.html#term-diis-start-convergence-dcft"><span class="xref std std-term">DIIS_START_CONVERGENCE</span></a> option to the value greater than
<img class="math" src="_images/math/0f69792f2e4e1a31c72973b66b73fecc4ae64145.png" alt="10^{-3}" style="vertical-align: -1px"/> by one or two orders of magnitude (e.g. <img class="math" src="_images/math/40d8abff1a6a8a7e56a4dedffa4b92b3c55db530.png" alt="10^{-2}" style="vertical-align: -1px"/> or
<img class="math" src="_images/math/5bf767368dd1ee6eb18d8fd7d7f4c2c301d5e35a.png" alt="10^{-1}" style="vertical-align: -1px"/>). This can be particularly useful for computions using the
ODC methods, because it can greatly reduce the number of iterations.</li>
<li>If oscillatory convergence is observed for atoms or molecules with high
symmetry, it is recommended to use the quadratically-convergent algorithm.</li>
<li>When using the quadratically-convergent algorithm for the closed-shell molecules, it
is recommended to set the <a class="reference internal" href="autodoc_glossary_options_c.html#term-qc-coupling-dcft"><span class="xref std std-term">QC_COUPLING</span></a> option to FALSE for efficiency
reasons (set by default).</li>
<li>For the ODC computations, the user has a choice of performing the computation of the guess orbitals and cumulants
using the corresponding DC method (set <a class="reference internal" href="autodoc_glossary_options_c.html#term-odc-guess-dcft"><span class="xref std std-term">ODC_GUESS</span></a> to TRUE). This can often lead to
significant computational savings, since the orbital update step in the DC methods is cheap.
Convergence of the guess orbitals and cumulants can be controlled using the
<a class="reference internal" href="autodoc_glossary_options_c.html#term-guess-r-convergence-dcft"><span class="xref std std-term">GUESS_R_CONVERGENCE</span></a> option.</li>
</ul>
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<h3><a href="index.html">Table Of Contents</a></h3>
<ul>
<li><a class="reference internal" href="#">DCFT: Density Cumulant Functional Theory</a><ul>
<li><a class="reference internal" href="#theory">Theory</a></li>
<li><a class="reference internal" href="#methods">Methods</a></li>
<li><a class="reference internal" href="#iterative-algorithms">Iterative Algorithms</a></li>
<li><a class="reference internal" href="#analytic-gradients">Analytic Gradients</a></li>
<li><a class="reference internal" href="#methods-summary">Methods Summary</a></li>
<li><a class="reference internal" href="#minimal-input">Minimal Input</a></li>
<li><a class="reference internal" href="#recommendations">Recommendations</a></li>
</ul>
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