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484 lines
19 KiB
TeX
484 lines
19 KiB
TeX
%
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% $Id$
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%
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\label{sec:mp2}
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\label{sec:rimp2}
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There are (at least) three algorithms within NWChem that compute the
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M{\o}ller-Plesset (or many-body) perturbation theory second-order
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correction to the Hartree-Fock energy (MP2). They vary in capability,
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the size of system that can be treated and use of other approximations
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\begin{itemize}
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\item Semi-direct --- this is recommended for most large applications
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(up to about 2800 basis functions), especially on the IBM SP and
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other machines with significant disk I/O capability. Partially
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transformed integrals are stored on disk, multi-passing as necessary.
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RHF and UHF references may be treated including computation of
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analytic derivatives. This is selected by specifying \verb+mp2+ on
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the task directive, e.g.
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\begin{verbatim}
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TASK MP2
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\end{verbatim}
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\item Fully-direct --- this is of utility if only limited I/O
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resources are available (up to about 2800 functions). Only RHF
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references and energies are available. This is selected by
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specifying \verb+direct_mp2+ on the task directive, e.g.
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\begin{verbatim}
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TASK DIRECT_MP2
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\end{verbatim}
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\item Resolution of the identity (RI) approximation MP2 (RI-MP2) ---
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this uses the RI approximation and is therefore only exact in the
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limit of a complete fitting basis. However, with some care, high
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accuracy may be obtained with relatively modest fitting basis sets.
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An RI-MP2 calculation can cost over 40 times less than the
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corresponding exact MP2 calculation. RHF and UHF references with
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only energies are available. This is selected by specifying
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\verb+rimp2+ on the task directive, e.g.,
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\begin{verbatim}
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TASK RIMP2
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\end{verbatim}
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\end{itemize}
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All three MP2 tasks share the same input block.
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\begin{verbatim}
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MP2
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[FREEZE [[core] (atomic || <integer nfzc default 0>)] \
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[virtual <integer nfzv default 0>]]
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[TIGHT]
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[PRINT]
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[NOPRINT]
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[VECTORS <string filename default scf-output-vectors> \
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[swap [(alpha||beta)] <integer pair-list>] ]
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[RIAPPROX <string riapprox default V>]
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[FILE3C <string filename default $file_prefix$.mo3cint">]
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[SCRATCHDISK <integer>]
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END
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\end{verbatim}
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\section{{\tt FREEZE} --- Freezing orbitals}
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\label{mp2:core}
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All MP2 modules support frozen core orbitals, however, only the direct
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MP2 and RI-MP2 modules support frozen virtual orbitals.
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By default, no orbitals are frozen. The \verb+atomic+ keyword causes
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orbitals to be frozen according to the rules in Table
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\ref{tbl:freeze-by-atoms}. Note that {\em no} orbitals are frozen on
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atoms on which the nuclear charge has been modified either by the user
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or due to the presence of an ECP. The actual input would be
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\begin{verbatim}
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freeze atomic
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\end{verbatim}
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For example, in a calculation on $Si(OH)_2$, by default the lowest
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seven orbitals would be frozen (the oxygen 1$s$, and the silicon 1$s$,
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2$s$ and 2$p$).
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\begin{table}[htbp]
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\label{tbl:freeze-by-atoms}
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\center
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\begin{tabular}{cclr}
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\hline\hline
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Period & Elements & Core Orbitals & Number of Core \\
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\hline
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0 & H -- He & --- & 0 \\
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1 & Li -- Ne & 1$s$ & 1 \\
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2 & Na -- Ar & 1$s$2$s$2$p$ & 5 \\
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3 & K -- Kr & 1$s$2$s$2$p$3$s$3$p$ & 9 \\
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4 & Rb -- Xe & 1$s$2$s$2$p$3$s$3$p$4$s$3$d$4$p$ & 18 \\
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5 & Cs -- Rn & 1$s$2$s$2$p$3$s$3$p$4$s$3$d$4$p$5$s$4$d$5$p$ & 27 \\
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6 & Fr -- Lr & 1$s$2$s$2$p$3$s$3$p$4$s$3$d$4$p$5$s$4$d$5$p$6$s$4$f$5$d$6$p$
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& 43 \\
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\hline\hline
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\end{tabular}
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\caption{Number of orbitals considered ``core'' in the ``freeze by
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atoms'' algorithm.}
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\end{table}
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{\em Caution:\/} The rule for freezing orbitals ``by atoms'' are
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rather unsophisticated: the number of orbitals to be frozen is
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computed from the Table \ref{tbl:freeze-by-atoms} by summing the number
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of core orbitals in each atom present. The corresponding number of
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lowest-energy orbitals are frozen --- if for some reason the actual
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core orbitals are not the lowest lying, then correct results will not
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be obtained. From limited experience, it seems that special attention
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should be paid to systems including third- and higher- period atoms.
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The user may also specify the number of orbitals to be frozen by atom.
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Following the $Si(OH)_2$ example, the user could specify
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\begin{verbatim}
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freeze atomic O 1 Si 3
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\end{verbatim}
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In this case only the lowest four orbitals would be frozen. If the user does
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not specify the orbitals by atom, the rules default to Table \ref{tbl:freeze-by-atoms}.
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{\em Caution:\/} The system does not check for a valid number of orbitals per
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atom. If the user specifies to freeze more orbitals then are available for
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the atom, the system will not catch the error. The user must specify a logical
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number of orbitals to be frozen for the atom.
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The \verb+FREEZE+ directive may also be used to specify the number of
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core orbitals to freeze. For instance, to freeze the first 10 orbitals
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\begin{verbatim}
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freeze 10
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\end{verbatim}
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or equivalently, using the optional keyword \verb+core+
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\begin{verbatim}
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freeze core 10
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\end{verbatim}
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Again, note that if the 10 orbitals to be frozen do not correspond to
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the first 10 orbitals, then the \verb+swap+ keyword of the
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\verb+VECTORS+ directive must be used to order the input orbitals
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correctly (Section \ref{sec:mp2vectors}).
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To freeze the highest virtual orbitals, use the \verb+virtual+
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keyword. For instance, to freeze the top 5 virtuals
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\begin{verbatim}
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freeze virtual 5
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\end{verbatim}
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Again, note that this only works for the direct-MP2 and RI-MP2 energy
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codes.
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\section{{\tt TIGHT} --- Increased precision}
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The \verb+TIGHT+ directive can be used to increase the precision
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in the MP2 energy and gradients.
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By default the MP2 gradient package should compute energies accurate
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to better than a micro-Hartree, and gradients accurate to about five
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decimal places (atomic units). However, if there is significant
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linear dependence in the basis set the precision might not be this
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good. Also, for computing very accurate geometries or numerical
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frequencies, greater precision may be desirable.
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This option increases the precision to which both the SCF (from
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$10^{-6}$ to $10^{-8}$) and CPHF (from $10^{-4}$ to $10^{-6}$) are
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solved, and also tightens thresholds for computation of the AO and MO
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integrals (from $10^{-9}$ to $10^{-11}$) within the MP2 code.
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\section{{\tt SCRATCHDISK} --- Limiting I/O usage}
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This directive -- used only in the semi-direct algorithm -- allows to
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limit the per process disk usage. Mandatory argument for this keyword
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is the maximum number of MBytes.
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For example, the following input line
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\begin{verbatim}
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scratchdisk 512
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\end{verbatim}
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puts an upper limit of 512 MBytes to the semi-direct MP2 usage of disk
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(again, on a per process base).
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\section{{\tt PRINT} and {\tt NOPRINT}}
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The standard print control options are recognized. The list of
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recognized names are given in Table \ref{tbl:mp2-printable}.
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\begin{table}
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\caption{Printable items in the MP2 modules and their default print levels.}
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\label{tbl:mp2-printable}
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\begin{tabular}{lll}
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\hline\hline
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Item & Print Level & Description \\
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\hline
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& & \\
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{\bf RI-MP2} & & \\ \hline
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& & \\
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``2/3 ints'' & debug & Partial 3-center integrals \\
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``3c ints'' & debug & MO 3-center integrals \\
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``4c ints b'' & debug & ``B'' matrix with approx. 4c integrals \\
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``4c ints'' & debug & Approximate 4-center integrals \\
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``amplitudes'' & debug & ``B'' matrix with denominators \\
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``basis'' & high & \\
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``fit xf'' & debug & Transformation for fitting basis \\
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``geombas'' & debug & Detailed basis map info\\
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``geometry'' & high & \\
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``information'' & low & General information about calc.\\
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``integral i/o'' & high & File size information\\
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``mo ints'' & debug & \\
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``pair energies'' & debug & (working only in direct\_mp2) \\
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``partial pair energies'' & debug & Pair energy matrix each time it is updated \\
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``progress reports'' & default & Report completion of time-consuming steps\\
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``reference'' & high & Details about reference wavefunction\\
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``warnings'' & low & Non-fatal warnings \\
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\hline\hline
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\end{tabular}
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\end{table}
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\section{{\tt VECTORS} --- MO vectors}
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\label{sec:mp2vectors}
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All of the (supported) MP2 modules require use of converged canonical
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SCF (RHF or UHF) orbitals for correct results. The vectors are by
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default obtained from the preceding SCF calculation, but it is
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possible to specify a different source using the \verb+VECTORS+
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directive. For instance, to obtain vectors from the file
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\verb+/tmp/h2o.movecs+, use the directive
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\begin{verbatim}
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vectors /tmp/h2o.movecs
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\end{verbatim}
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As noted above (Section \ref{mp2:core}) if the SCF orbitals are not in
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the correct order, it is necessary to permute the input orbitals using
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the \verb+swap+ keyword of the \verb+VECTORS+ directive. For
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instance, if it is desired to freeze a total six orbitals
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corresponding to the SCF orbitals 1--5, and 7, it is necessary to swap
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orbital 7 into the 6th position. This is accomplished by
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\begin{verbatim}
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vectors swap 6 7
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\end{verbatim}
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The swap capability is examined in more detail in Section
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\ref{sec:vectors}.
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\section{RI-MP2 fitting basis}
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\sloppy
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The RI-MP2 method requires a fitting basis, which must be specified
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with the name \verb+"ri-mp2 basis"+ (see Section \ref{sec:basis}).
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For instance,
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\begin{verbatim}
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basis "ri-mp2 basis"
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O s; 10000.0 1
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O s; 1000.0 1
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O s; 100.0 1
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...
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end
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\end{verbatim}
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Alternatively, using a standard capability of basis sets (Section
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\ref{sec:basis}) another named basis may be associated with the
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fitting basis. For instance, the following input specifies a basis
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with the name \verb+"small fitting basis"+ and then defines this to be
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the \verb+"ri-mp2 basis"+.
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\begin{verbatim}
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basis "small fitting basis"
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H s; 10 1
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H s; 3 1
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H s; 1 1
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H s; 0.1 1
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H s; 0.01 1
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end
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set "ri-mp2 basis" "small fitting basis"
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\end{verbatim}
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\fussy
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\section{{\tt FILE3C} --- RI-MP2 3-center integral filename}
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\sloppy
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The default name for the file used to store the transformed 3-center
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integrals is \verb+"$file_prefix$.mo3cint"+ in the scratch directory.
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This may be overridden using the FILE3C directive. For instance, to
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specify the file \verb+/scratch/h2o.3c+, use this directive
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\begin{verbatim}
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file3c /scratch/h2o.3c
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\end{verbatim}
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\fussy
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\section{{\tt RIAPPROX} --- RI-MP2 Approximation}
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The type of RI approximation used in the RI-MP2 calculation is controlled
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by means of the RIAPPROX directive. The two possible values are
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\verb+V+ and \verb+SVS+ (case sensitive), which correspond to the
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approximations with the same names described in O.~Vahtras, J~Alml\"of,
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and M.~W.~Feyereisen, {\em Chem. Phys. Lett.} {\bf 213}, 514--518
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(1993). The default is \verb+V+.
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% The \verb+"S"+ approximation will also be supported eventually.
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\section{Advanced options for RI-MP2}
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These options, which functioned at the time of writing, are not
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currently supported.
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\subsection{Control of linear dependence}
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Construction of the RI fit requires the inversion of a matrix of
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fitting basis integrals which is carried out via diagonalization. If
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the fitting basis includes near linear dependencies, there will be
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small eigenvalues which can ultimately lead to non-physical RI-MP2
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correlation energies. Eigenvectors of the fitting matrix
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are discarded if the corresponding eigenvalue is less than
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\verb+$mineval$+ which defaults to $10^{-8}$. This
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parameter may be changed by setting the a parameter in the database.
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For instance, to set it to $10^{-10}$
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\begin{verbatim}
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set "mp2:fit min eval" 1e-10
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\end{verbatim}
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\subsection{Reference Spin Mapping for RI-MP2 Calculations}
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The user has the option of specifying that the RI-MP2 calculations are
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to be done with variations of the SCF reference wavefunction. This is
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accomplished with a \verb+SET+ directive of the form,
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\begin{verbatim}
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set "mp2:reference spin mapping" <integer array default 0>
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\end{verbatim}
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Each element specified for \verb+array+ is the SCF spin case to be
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used for the corresponding spin case of the correlated calculation.
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The number of elements set determines the overall type of correlated
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calculation to be performed. The default is to use the unadulterated
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SCF reference wavefunction.
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For example, to perform a spin-unrestricted calculation (two elements)
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using the alpha spin orbitals (spin case 1) from the reference for
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both of the correlated reference spin cases, the \verb+SET+ directive
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would be as follows,
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\begin{verbatim}
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set "mp2:reference spin mapping" 1 1
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\end{verbatim}
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The SCF calculation to produce the reference wavefunction could be either
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RHF or UHF in this case.
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The \verb+SET+ directive for a similar case, but this time using the
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beta-spin SCF orbitals for both correlated spin cases, is as follows,
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\begin{verbatim}
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set "mp2:reference spin mapping" 2 2
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\end{verbatim}
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The SCF reference calculation must be UHF in this case.
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The \verb+SET+ directive for a spin-restricted calculation (one
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element) from the beta-spin SCF orbitals using this option is as
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follows,
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\begin{verbatim}
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set "mp2:reference spin mapping" 2
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\end{verbatim}
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The \verb+SET+ directive for a spin-unrestricted calculation with the
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spins flipped from the original SCF reference wavefunction is as
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follows,
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\begin{verbatim}
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set "mp2:reference spin mapping" 2 1
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\end{verbatim}
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\subsection{Batch Sizes for the RI-MP2 Calculation}
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The user can control the size of each batch in the transformation and
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energy evaluation in the MP2 calculation, and consequently the memory
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requirements and number of passes required. This is done using two
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\verb+SET+ directives of the following form,
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\begin{verbatim}
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set "mp2:transformation batch size" <integer size default -1>
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set "mp2:energy batch size" <integer isize jsize default -1 -1>
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\end{verbatim}
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The default is for the code to determine the batch size based on the
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available memory. Should there be problems with the
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program-determined batch sizes, these variables allow the user to
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override them. The program will always use the smaller of the user's
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value of these entries and the internally computed batch size.
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The transformation batch size computed in the code is the number of
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occupied orbitals in the $({occ}\ {vir} | {fit})$ three-center
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integrals to be produced at a time. If this entry is less than the
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number of occupied orbitals in the system, the transformation will
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require multiple passes through the two-electron integrals. The
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memory requirements of this stage are {\em two} global arrays of
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dimension ${<batch size>}\times {vir} \times {fit}$ with the ``fit''
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dimension distributed across all processors (on shell-block
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boundaries). The compromise here is memory space versus multiple
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integral evaluations.
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The energy evaluation batch sizes are computed in the code from the
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number of occupied orbitals in the two sets of three-center
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integrals to be multiplied together to produce a matrix of approximate
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four-center integrals. Two blocks of integrals of dimension $({<batch
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isize>}\times {vir})$ and $({<batch jsize>}\times {vir})$ by fit are
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read in from disk and multiplied together to produce $<batch isize>
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<batch jsize> {vir}^2$ approximate integrals. The compromise here is
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performance of the distributed matrix multiplication (which requires
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large matrices) versus memory space.
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\subsection{Energy Memory Allocation Mode: RI-MP2 Calculation}
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The user must choose a strategy for the memory allocation in the energy
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evaluation phase of the RI-MP2 calculation, either by minimizing the amount
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of I/O, or minimizing the amount of computation. This can be accomplished
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using a \verb+SET+ directive of the form,
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\begin{verbatim}
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set "mp2:energy mem minimize" <string mem_opt default I>
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\end{verbatim}
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A value of \verb+I+ entered for the string \verb+mem_opt+ means that a
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strategy to minimize I/O will be employed. A value of \verb+C+ tells
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the code to use a strategy that minimizes computation.
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When the option to minimize I/O is selected, the block sizes are made
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as large as possible so that the total number of passes through the
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integral files is as small as possible. When the option to minimize
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computation is selected, the blocks are chosen as close to square as
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possible so that permutational symmetry in the energy evaluation can
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be used most effectively.
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\subsection{Local Memory Usage in Three-Center Transformation}
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For most applications, the code will be able to size the blocks
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without help from the user. Therefore, it is unlikely that users will
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have any reason to specify values for these entries except when doing
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very particular performance measurements.
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The size of \verb+xf3ci:AO 1 batch size+ is the most important of the
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three, in terms of the effect on performance.
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Local memory usage in the first two steps of the transformation is
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controlled in the RI-MP2 calculation using the following \verb+SET+
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directives,
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\begin{verbatim}
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set "xf3ci:AO 1 batch size" <integer max>
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set "xf3ci:AO 2 batch size" <integer max>
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set "xf3ci:fit batch size" <integer max>
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\end{verbatim}
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The size of the local arrays determines the sizes of the two matrix
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multiplications. These entries set limits on the size of blocks to be
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used in each index. The listing above is in order of importance of
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the parameters to performance, with \verb+xf3ci:AO 1 batch size+ being
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most important.
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Note that these entries are only upper bounds and that the program
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will size the blocks according to what it determines as the best usage of
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the available local memory. The absolute maximum for a block size is
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the number of functions in the AO basis, or the number of fitting basis
|
|
functions on a node. The absolute minimum value for block size is the
|
|
size of the largest shell in the appropriate basis. Batch size entries
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|
specified for \verb+max+ that are larger than these limits are
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|
automatically reset to an appropriate value.
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\section{One-electron properties and natural orbitals}
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If an MP2 energy gradient is computed, all contributions are available
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|
to form the MP2 linear-response density. This is the density that
|
|
when contracted with any spin-free, one-electron operator yields the
|
|
associated property defined as the derivative of the energy. Thus,
|
|
the reported MP2 dipole moment is the derivative of the energy
|
|
w.r.t. an external magnetic field and is {\em not} the expectation
|
|
value of the operator over the wavefunction. Only dipole moments are
|
|
printed by the MP2 gradient code, but natural orbitals are produced
|
|
and stored in the permanent directory with a file extension of
|
|
\verb+".mp2nos"+. These may be fed into the property package (see
|
|
Section \ref{sec:property}) to compute more general properties. Note
|
|
that the MP2 linear response density matrix is not necessarily
|
|
positive definite so it is not unusual to see a few small negative
|
|
natural orbital occupation numbers.
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