Diagrammatic Formulation of MPPT#

MPPT builds a second-order effective Hamiltonian in an explicit, possibly incomplete determinant model space. Its diagrammatic formulation organizes the contributions from external excitations while retaining the partitioning associated with each model determinant.

The MPPT Construction#

Diagonalize → perturb → diagonalize. First solve the variational problem in the model space P. Use auxiliary model-space states to construct the denominators, assemble the perturbative correction, then diagonalize the corrected Hamiltonian. The last step updates the model-space coefficients and state mixing, not just the energies.

Model States, Determinants And Projectors#

Let N_P be the number of orthonormal determinants retained in the model space. The space and its projector are different objects:

\[\begin{split}\begin{aligned} \mathcal P &= \operatorname{span}\{|J\rangle\}_{J=1}^{N_P},\\ P &= \sum_{J=1}^{N_P}|J\rangle\langle J|,\\ Q &= 1-P. \end{aligned}\end{split}\]

The range of Q is the external space, denoted by calligraphic Q below. A selected model space need not contain every determinant of a CAS.

The homepage uses independent state indices μ and ν, and independent determinant indices I and J:

\[\begin{split}\begin{aligned} \langle\Psi_\mu^0| &= \sum_{I=1}^{N_P}C_{I\mu}^*\langle I|,\\ |\Psi_\nu^0\rangle &= \sum_{J=1}^{N_P}C_{J\nu}|J\rangle. \end{aligned}\end{split}\]

Independent indices allow both diagonal and off-diagonal matrix elements; they do not impose unequal values. Using μ on both sides specifies a same-state expectation value. Reusing J as a dummy index in two separate sums is legal, but I and J make their independence explicit when the sums are combined.

Writing J as a member of P is conventional shorthand only if P also denotes the determinant index set. Explicit bounds avoid conflating an integer label, a determinant vector, a subspace and its projector. Hole and particle labels belong to orbital substitutions, not to these model-state expansions.

Determinant-Specific Partitionings#

Each model determinant J defines a partitioning of the same Hamiltonian:

\[\begin{split}\begin{aligned} H &= H_0(J) + V(J),\\ H_0(J)|J\rangle &= e_J|J\rangle,\\ D_{JA} &= e_J - E_A^{(0)}(J). \end{aligned}\end{split}\]

The determinant labelled A belongs to the external space. Both energies in its denominator use the partitioning of J; the external determinant does not introduce a new partitioning. The external space can include omitted active-space determinants as well as excitations outside that CAS.

Density Matrices And IP/EA-Like Energies#

The normalized auxiliary states above supply density matrices and associated Fock-like matrices that define separate electron-removal and electron-attachment orbital energies. For a spin orbital v with occupation n, the unrelaxed definitions are:

\[\begin{split}\begin{aligned} E_\mu &= \langle\Psi_\mu^0|H|\Psi_\mu^0\rangle,\\ n_{\mu v} &= \langle\Psi_\mu^0|a_v^\dagger a_v|\Psi_\mu^0\rangle,\\ \varepsilon_{\mu v}^{\oplus} &= E_\mu - \frac{\langle\Psi_\mu^0|a_v^\dagger H a_v|\Psi_\mu^0\rangle}{n_{\mu v}},\\ \varepsilon_{\mu v}^{\odot} &= \frac{\langle\Psi_\mu^0|a_v H a_v^\dagger|\Psi_\mu^0\rangle}{1-n_{\mu v}} - E_\mu. \end{aligned}\end{split}\]

The removal expression requires nonzero occupation; the attachment expression requires nonzero vacancy. Their signs correspond to minus the ionization potential and minus the electron affinity, respectively. For several auxiliary states, removal energies are averaged with occupation weights and attachment energies with vacancy weights. In the state-averaged implementation, these are diagonal ratios of the corresponding Fock-like and density matrices.

This is an IP/EA-based orbital-energy construction, rather than the empirical IPEA shift used in CASPT2. Which orbital is occupied depends on J, giving distinct partitionings even when the orbital-energy sets are shared. The spin-adapted form includes the occupation-dependent two-body term needed for singly occupied spatial orbitals; see the 1997 and 1999 papers below.

Corrected Effective Hamiltonian#

The matrices are added in the same basis. W⁽²⁾ denotes the second-order correction in the Hermitian MPPT formulation:

\[H_{\mathrm{eff}}^{[2]} = PHP + W^{(2)}.\]

The superscript (2) denotes the contribution of exactly second order; [2] denotes the Hamiltonian assembled through that order. PHP already contains the full Hamiltonian within the model space. It is not one of the determinant-specific zeroth-order operators H₀(J).

The correction comes from coupling to an external intermediate and back. The right-partitioned expression uses the denominator associated with the ket determinant J:

\[K_{IJ}^{(2)} = \sum_{|A\rangle\in\mathcal Q} \frac{\langle I|H|A\rangle\langle A|H|J\rangle}{D_{JA}}.\]

For real, nonzero denominators and Hermitian H, the Hermitian second-order form is:

\[\begin{split}\begin{aligned} W^{(2)} &= \tfrac12\bigl(K^{(2)}+K^{(2)\dagger}\bigr),\\ W_{IJ}^{(2)} &= \frac12\sum_{|A\rangle\in\mathcal Q} \langle I|H|A\rangle\langle A|H|J\rangle\\ &\qquad\times\left(\frac1{D_{IA}}+\frac1{D_{JA}}\right). \end{aligned}\end{split}\]

W is an effective-Hamiltonian correction, not the perturbation V(J) itself and not a wave-function correction. Two model–external couplings make this a second-order term. All model-space intermediates are excluded from the sum. These equations are unshifted; numerical level shifts are separate settings.

Why Diagonalize Twice?#

  1. Diagonalize PHP. Obtain variational model-space states and use the chosen auxiliary states to construct densities, orbital energies and denominators.

  2. Build W⁽²⁾. Evaluate both diagonal and off-diagonal corrections through external intermediates, then add them to PHP in the determinant basis.

  3. Diagonalize the assembled matrix. Obtain corrected energies and new model-space coefficients. This updates state mixing at fixed orbitals.

In a full eigenbasis of PHP, the same matrix assembly reads:

\[\begin{split}\begin{aligned} H_{\mu\nu}^{\mathrm{aux},[2]} &= E_\mu^{\mathrm{var}}\delta_{\mu\nu}+W_{\mu\nu}^{\mathrm{aux},(2)},\\ W_{\mu\nu}^{\mathrm{aux},(2)} &= \sum_{I,J=1}^{N_P}C_{I\mu}^*W_{IJ}^{(2)}C_{J\nu}. \end{aligned}\end{split}\]

The corrected energies are the eigenvalues of this full matrix. Adding only one scalar correction to each variational energy misses the off-diagonal mixing that matters for quasi-degenerate states. Rediagonalizing also does not mean adding the eigenvalues of two separately diagonalized matrices.

In the DIAGPT HDF5 handoff, h1_packed stores the model-space matrix and h2_packed stores the already assembled second-order effective Hamiltonian, with the reference-energy offset stored separately. h2_packed is not W⁽²⁾: adding h1_packed to it would count the model-space matrix twice.

The Closed Diagram On The Homepage#

The original ASCII stars and horizontal dashes are retained, with orbital labels i, a, b and j and no arrow overlays. Its double-loop topology illustrates a diagonal second-order double-excitation contribution. The horizontal lines represent the two-electron interactions, with the lower interaction preceding the upper one.

The external intermediate is generated by removing electrons from spin orbitals i and j and placing them in a and b:

\[|A_{ij}^{ab}(J)\rangle = a_a^\dagger a_b^\dagger a_j a_i|J\rangle \in\mathcal Q.\]

Indices i and j are occupied in J; a and b are empty. These orbital indices are distinct from the determinant labels I and J and the state labels μ and ν. The full diagonal double-substitution contribution is:

\[\begin{split}W_{JJ,\mathrm{double}}^{(2)} = \sum_{\substack{i<j,\ a<b\\|A_{ij}^{ab}(J)\rangle\in\mathcal Q}} \frac{|\langle A_{ij}^{ab}(J)|H|J\rangle|^2} {D_{J,A_{ij}^{ab}(J)}}.\end{split}\]

Each external determinant is counted once. An unrestricted ordered sum over both pairs would instead require a factor of ¼. The full correction also includes single substitutions and off-diagonal determinant couplings; the closed drawing does not represent all those terms.

For the closed contribution, the bra and ket determinant coincide. This does not require the model-state indices μ and ν to coincide. A diagonal determinant contribution with value w_J transforms as:

\[\begin{split}\begin{aligned} W_{IJ}^{\mathrm{closed}} &= \delta_{IJ}w_J,\\ W_{\mu\nu}^{\mathrm{closed,aux}} &= \sum_{J=1}^{N_P}C_{J\mu}^*w_J C_{J\nu}. \end{aligned}\end{split}\]

It can therefore contribute to off-diagonal elements in the model-state basis. The general bra/ket expansions on the homepage describe that change of basis; they do not turn the closed drawing into a general off-diagonal determinant diagram.

Denominator contours encode the reduced resolvent. They are distinct from the fermion contractions: when working with a product of two matrix elements, Wick contractions are taken inside each expectation value.

Relativistic Contributions#

GRECP and GRPP name the same generalized relativistic pseudopotential framework. It represents scalar-relativistic and spin–orbit interactions while replacing core electrons. Spin–orbit coupling is included in the effective-Hamiltonian calculation of the final levels.

Gatchina parameter sets constructed for that purpose can also include Breit and QED effects — electron self-energy and vacuum polarization. These effects are inherited from the chosen pseudopotential; they are not generated by turning on a universal QED switch in MPPT.

The Gatchina pseudopotential database supplies parameter sets and basis information. LIBGRPP evaluates their molecular integrals. For QED-containing constructions, see Zaitsevskii et al., Generalized relativistic small-core pseudopotentials accounting for quantum electrodynamic effects: construction and pilot applications, open text, DOI.

Recent Application#

M. M. Seregin, I. O. Glebov, N. S. Mosyagin, A. V. Oleynichenko and A. V. Zaitsevskii, A Quasirelativistic Perturbation Theory As a Tool for Modeling f–f and f–d Transitions in Lanthanide Ions, Russian Journal of Physical Chemistry A 100, 696–705 (2026). Published article · ResearchGate.

Conceptual Line#

time-ordered QED diagrams
-> linked many-body perturbation diagrams
-> multiconfigurational model-space diagrams
-> Hose-Kaldor general-model-space diagrams
-> Zaitsevskii-Cimiraglia multipartitioning diagrams
-> second-order MPPT effective Hamiltonian
-> modern effective-Hamiltonian and active-space descendants

The important continuity is algebraic rather than visual. The diagram language corresponds to projectors, resolvents, effective Hamiltonians, connected contractions, and internal/external-space partitions in modern implementations.

Foundational Papers#

Year

Source

DOI

1957

Goldstone, Derivation of the Brueckner many-body theory

https://doi.org/10.1098/rspa.1957.0037

1974

Lindgren, The Rayleigh-Schrodinger perturbation and the linked-diagram theorem for a multi-configurational model space

https://doi.org/10.1088/0022-3700/7/18/010

1979

Hose and Kaldor, Diagrammatic many-body perturbation theory for general model spaces

https://doi.org/10.1088/0022-3700/12/23/012

1980

Hose and Kaldor, A General-Model-Space Diagrammatic Perturbation Theory

https://doi.org/10.1088/0031-8949/21/3-4/019

1981

Hose and Kaldor, The shifted scheme in the general-model-space diagrammatic perturbation theory

https://doi.org/10.1016/0301-0104(81)85140-3

1982

Hose and Kaldor, Quasidegenerate perturbation theory

https://doi.org/10.1021/j100209a005

1985

Cimiraglia, Second order perturbation correction to CI energies by use of diagrammatic techniques

https://doi.org/10.1063/1.449362

1995

Zaitsevskii and Malrieu, Multi-partitioning quasidegenerate perturbation theory

https://doi.org/10.1016/0009-2614(94)01503-N

1996

Cimiraglia, Many-body multireference Moller-Plesset and Epstein-Nesbet perturbation theory

https://doi.org/10.1002/(SICI)1097-461X(1996)60:1<167::AID-QUA18>3.0.CO;2-C

1997

Zaitsevskii and Malrieu, Spin-adapted multipartitioning perturbation theory

https://doi.org/10.1007/s002140050230

1999

Zaitsevskii and Cimiraglia, Diagrammatic formulation of the second-order many-body multipartitioning perturbation theory

https://doi.org/10.1002/(SICI)1097-461X(1999)73:5<395::AID-QUA2>3.0.CO;2-T

Reading The Diagrams#

Diagram idea

Meaning in MPPT

Model space

determinant/configuration subspace used to build the effective Hamiltonian

External lines

hole/particle differences between reference and external determinants

Denominator contour

resolvent or energy denominator for a perturbative contribution

Linked topology

connected contribution retained in the second-order kernel

Diagram class

contributions grouped by the occupation differences between model determinants

Bra/ket partitioning

determinant-specific zeroth-order operators and denominator choices

Modern Descendants#

Foundational idea

Contemporary family

Representative source

General-model-space effective Hamiltonian

Intermediate-Hamiltonian Fock-space coupled cluster

Meissner 2001, https://doi.org/10.1016/S0065-3276(05)39011-3

Determinant-selected model spaces

Selected-space MRPT

Malrieu and Heully 2023, https://doi.org/10.1063/5.0153416

Active/reference-space plus second-order external correction

CASPT2, NEVPT2, GASPT2, XMS-CASPT2

Ma et al. 2016, https://doi.org/10.1021/acs.jctc.6b00382

Effective-Hamiltonian reduction of large excited-state spaces

DFT/MRCI(2), QDPT solvers

Neville and Schuurman 2022, https://doi.org/10.1063/5.0118285

Diagrammatic response theory

ADC / intermediate-state representation

Dreuw et al. 2023, https://doi.org/10.1021/acs.jpca.3c02761

Renormalized multireference perturbation theory

DSRG-MRPT2/3

Zhao and Evangelista 2024, https://doi.org/10.1021/acs.jpclett.4c01372

Practical Reading Order#

  1. Hose and Kaldor 1979 for the general-model-space diagram language.

  2. Zaitsevskii and Cimiraglia 1999 for the second-order multipartitioning diagram formulation.

  3. Cimiraglia 1985 and 1996 for the determinant-based second-order correction path connected to CIPSI and Epstein-Nesbet partitioning.

  4. Malrieu and Heully 2023 for the closest modern update to selected determinant model spaces.

  5. Meissner 2001 for the bridge from model-space perturbation theory to intermediate-Hamiltonian coupled-cluster methods.