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The one-electron reduced density matrix (1RDM) contains information useful for the prediction of molecular properties (for instance, dipole moments). As an example, the 1RDM can be easily obtained in pySCF for the $\ce{H2O}$ by:

import pyscf

mol = pyscf.M(
    atom = 'H 0 0 0; F 0 0 1.1',
    basis = 'ccpvdz')

mf = mol.HF().run()
mycc = mf.CISD().run()

dm1 = mycc.make_rdm1()

giving:

array([[ 1.99994077e+00, -1.99162245e-04, -7.87231222e-05,
        -9.53691579e-19, -9.50673436e-20, -2.30038238e-05,
         1.84071578e-05, -2.70900398e-17,  1.47460417e-16,
        -1.60886286e-04, -1.14196814e-16, -1.91359638e-16,
         1.67836021e-04, -2.20982363e-04, -4.94641186e-19,
         6.25007588e-18,  5.02802627e-18, -1.03628178e-17,
         1.11486983e-05],
       [-1.99162245e-04,  1.98848063e+00,  6.13179991e-03,
         4.08139213e-18, -9.19419529e-19,  1.76764571e-03,
        -2.80098013e-03,  2.23749372e-16,  1.96829216e-17,
         6.47137404e-03,  1.34300623e-16,  8.07794933e-18,
         4.05861098e-03, -3.70627178e-04, -8.38703764e-19,
         5.35827360e-18, -9.79839661e-18,  4.01319679e-19,
         4.31669973e-04],
       [-7.87231222e-05,  6.13179991e-03,  1.96133369e+00,
        -1.06540989e-17,  3.71419450e-18, -2.13975050e-02,
        -1.34538561e-02,  1.24728207e-17, -3.65798882e-16,
        -1.07773571e-02, -7.72135156e-17, -2.80257053e-16,
        -1.70765070e-02, -2.41959181e-03,  3.03116775e-18,
         5.42133662e-19,  7.04961364e-20, -1.81295382e-17,
         2.56630622e-03],
       [-9.53691579e-19,  4.08139213e-18, -1.06540989e-17,
         1.98087586e+00, -2.42526709e-18, -4.00633524e-16,
         6.82988337e-17,  1.45808332e-03, -7.75567418e-03,
         2.02353702e-17, -3.90555928e-04, -2.27416768e-03,
         3.59002056e-19,  1.15318072e-16, -2.66714325e-17,
         3.79612632e-16,  2.00404079e-04,  4.35714707e-04,
        -1.76355694e-17],
       [-9.50673436e-20, -9.19419529e-19,  3.71419450e-18,
        -2.42526709e-18,  1.98087586e+00,  7.68379095e-17,
         2.96383736e-17, -7.75567418e-03, -1.45808332e-03,
         2.03613686e-18, -2.27416768e-03,  3.90555928e-04,
         3.18752574e-17,  8.58758188e-18,  1.32729655e-17,
         1.15457073e-17,  4.35714707e-04, -2.00404079e-04,
         1.01725219e-17],
       [-2.30038238e-05,  1.76764571e-03, -2.13975050e-02,
        -4.00633524e-16,  7.68379095e-17,  1.84368423e-02,
         1.45015635e-02, -5.82749480e-18, -1.19950717e-17,
        -5.16805460e-03, -4.05467609e-18, -3.32237435e-18,
        -7.52408964e-03, -7.56030680e-04,  2.06258966e-19,
        -6.79531535e-19,  2.36816792e-19,  1.29010476e-20,
        -1.38189704e-05],
       [ 1.84071578e-05, -2.80098013e-03, -1.34538561e-02,
         6.82988337e-17,  2.96383736e-17,  1.45015635e-02,
         1.25647125e-02, -5.04136605e-18, -3.79435707e-18,
        -4.41175502e-03, -4.92110092e-18, -7.88552981e-18,
        -5.56377189e-03, -2.23572205e-03,  3.62979262e-19,
        -9.42389633e-19,  8.25935435e-19,  1.18034368e-18,
         3.96084024e-04],
       [-2.70900398e-17,  2.23749372e-16,  1.24728207e-17,
         1.45808332e-03, -7.75567418e-03, -5.82749480e-18,
        -5.04136605e-18,  1.00099863e-02,  2.38122851e-17,
        -2.52114787e-18, -5.12613672e-03,  1.90552589e-03,
         2.95515608e-18,  1.00932446e-18, -8.80853860e-18,
         2.26350025e-18,  7.41491643e-04, -5.25910649e-04,
         4.53998014e-19],
       [ 1.47460417e-16,  1.96829216e-17, -3.65798882e-16,
        -7.75567418e-03, -1.45808332e-03, -1.19950717e-17,
        -3.79435707e-18,  2.38122851e-17,  1.00099863e-02,
         1.48963223e-17, -1.90552589e-03, -5.12613672e-03,
         9.27272866e-18, -1.90427871e-19,  1.78058821e-18,
        -3.70031467e-18,  5.25910649e-04,  7.41491643e-04,
         1.86549196e-19],
       [-1.60886286e-04,  6.47137404e-03, -1.07773571e-02,
         2.02353702e-17,  2.03613686e-18, -5.16805460e-03,
        -4.41175502e-03, -2.52114787e-18,  1.48963223e-17,
         6.93767668e-03,  2.20079534e-18, -1.64256033e-18,
         1.03536996e-03,  2.60881775e-03,  7.83668113e-20,
        -1.47994181e-18,  3.62037203e-19,  1.58980841e-18,
         1.11321275e-03],
       [-1.14196814e-16,  1.34300623e-16, -7.72135156e-17,
        -3.90555928e-04, -2.27416768e-03, -4.05467609e-18,
        -4.92110092e-18, -5.12613672e-03, -1.90552589e-03,
         2.20079534e-18,  4.45577120e-03,  1.51402375e-17,
         3.46729881e-18, -5.01502863e-19, -2.15430415e-18,
         4.61799403e-18,  7.25292075e-04, -1.93732359e-04,
        -7.52894088e-19],
       [-1.91359638e-16,  8.07794933e-18, -2.80257053e-16,
        -2.27416768e-03,  3.90555928e-04, -3.32237435e-18,
        -7.88552981e-18,  1.90552589e-03, -5.12613672e-03,
        -1.64256033e-18,  1.51402375e-17,  4.45577120e-03,
         6.08876803e-18,  2.82738131e-18,  2.46975026e-18,
         6.50590972e-18,  1.93732359e-04,  7.25292075e-04,
        -7.55165672e-19],
       [ 1.67836021e-04,  4.05861098e-03, -1.70765070e-02,
         3.59002056e-19,  3.18752574e-17, -7.52408964e-03,
        -5.56377189e-03,  2.95515608e-18,  9.27272866e-18,
         1.03536996e-03,  3.46729881e-18,  6.08876803e-18,
         5.26811829e-03, -1.01060468e-03, -2.64539985e-19,
         1.60654807e-18, -1.37431011e-19,  8.03483700e-19,
        -5.26829940e-04],
       [-2.20982363e-04, -3.70627178e-04, -2.41959181e-03,
         1.15318072e-16,  8.58758188e-18, -7.56030680e-04,
        -2.23572205e-03,  1.00932446e-18, -1.90427871e-19,
         2.60881775e-03, -5.01502863e-19,  2.82738131e-18,
        -1.01060468e-03,  3.76975815e-03, -1.88963598e-19,
        -4.45613527e-20,  3.86855826e-19,  5.15586386e-19,
        -7.78185559e-04],
       [-4.94641186e-19, -8.38703764e-19,  3.03116775e-18,
        -2.66714325e-17,  1.32729655e-17,  2.06258966e-19,
         3.62979262e-19, -8.80853860e-18,  1.78058821e-18,
         7.83668113e-20, -2.15430415e-18,  2.46975026e-18,
        -2.64539985e-19, -1.88963598e-19,  2.88726905e-03,
         3.41776421e-10,  1.32473979e-18, -1.84940088e-18,
         3.59982084e-19],
       [ 6.25007588e-18,  5.35827360e-18,  5.42133662e-19,
         3.79612632e-16,  1.15457073e-17, -6.79531535e-19,
        -9.42389633e-19,  2.26350025e-18, -3.70031467e-18,
        -1.47994181e-18,  4.61799403e-18,  6.50590972e-18,
         1.60654807e-18, -4.45613527e-20,  3.41776421e-10,
         2.88726834e-03, -2.09437953e-18, -1.29027293e-18,
        -1.79752988e-18],
       [ 5.02802627e-18, -9.79839661e-18,  7.04961364e-20,
         2.00404079e-04,  4.35714707e-04,  2.36816792e-19,
         8.25935435e-19,  7.41491643e-04,  5.25910649e-04,
         3.62037203e-19,  7.25292075e-04,  1.93732359e-04,
        -1.37431011e-19,  3.86855826e-19,  1.32473979e-18,
        -2.09437953e-18,  2.55700661e-03,  1.80635235e-18,
         4.73014873e-20],
       [-1.03628178e-17,  4.01319679e-19, -1.81295382e-17,
         4.35714707e-04, -2.00404079e-04,  1.29010476e-20,
         1.18034368e-18, -5.25910649e-04,  7.41491643e-04,
         1.58980841e-18, -1.93732359e-04,  7.25292075e-04,
         8.03483700e-19,  5.15586386e-19, -1.84940088e-18,
        -1.29027293e-18,  1.80635235e-18,  2.55700661e-03,
         2.13939315e-19],
       [ 1.11486983e-05,  4.31669973e-04,  2.56630622e-03,
        -1.76355694e-17,  1.01725219e-17, -1.38189704e-05,
         3.96084024e-04,  4.53998014e-19,  1.86549196e-19,
         1.11321275e-03, -7.52894088e-19, -7.55165672e-19,
        -5.26829940e-04, -7.78185559e-04,  3.59982084e-19,
        -1.79752988e-18,  4.73014873e-20,  2.13939315e-19,
         1.69602024e-03]])

I am interested in how to derive properties from the 1RDM and if there are any open source codes that can be used to calculate properties in this way. For instance, how does one obtain the dipole moment.

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  • 4
    $\begingroup$ All expectation values of one-body operators. Further, due to Gilbert's theorem (an analog to the Hohenberg-Kohn theorem), the ground state 1-RDM is in one-to-one correspondence with the ground state; hence, in principle, you can obtain all ground state properties of the system (of course, if all requirements of the theorem are satisfied). There are also constrained-search formulations of the ground state energy, similarly to the density (DFT) case. $\endgroup$
    – Jakob
    Jul 11, 2022 at 10:36
  • $\begingroup$ @Jakob This comment would probably work as an answer. $\endgroup$
    – Tyberius
    Jul 14, 2022 at 13:43
  • $\begingroup$ @Tyberius Well, I don't have a clue about the codes the OP asked for... $\endgroup$
    – Jakob
    Jul 14, 2022 at 13:56
  • 2
    $\begingroup$ @Jakob This is part of why we generally encourage users to keep their posts to one specific question, as asking multipart questions can discourage people from answering if they only know one part. I think you could provide an answer assuming the title is the question for this post. The details about code that implement this could be moved to a new question or someone could provide a separate answer here. $\endgroup$
    – Tyberius
    Jul 14, 2022 at 14:31

1 Answer 1

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I (think that I) can only help with the part in the title of the question. The following answer won't be mathematical rigorous and nothing will be proved, but references are provided at the end.

Notation: Let $ \mathfrak h$ denote a (complex) single-particle Hilbert space and $H_N:=\wedge^N \mathfrak h$ the corresponding (complex) Hilbert space of $N$ identical fermions$^1$. We denote by $\psi \in H_N$ a normalized $N$-particle state, the inner product on a complex Hilbert space $h$ by $\langle \cdot,\cdot\rangle_{h}$ and the trace as $\mathrm{Tr}_h$. The inner product is defined to be anti-linear in the first and linear in the second argument. We'll also work in second quantization, which however is just a matter of taste. To this end, we use the creation and annihilation operators $a^\dagger(f)$ and $a(f)$ for $f\in \mathfrak h$ as usual. For a self-adjoint operator $o$ on $\mathfrak h$, the corresponding one-body operator on $H_N$ is written as $O$. As a last point, we will use $\{f_j\}_{j\in J}$ to denote an orthonormal and complete basis on $\mathfrak h$.


For $\psi \in H_N$ and $f,g\in \mathfrak h$, we define the one-body reduced density matrix (1-RDM) by the relation $$ \langle f,\gamma_\psi g\rangle_{\mathfrak h} := \langle \psi, a^\dagger(g)\, a(f)\,\psi\rangle_{H_N}\quad.\tag{1}$$ By construction, $\gamma_\psi$ is positive semi-definite and trace-class (and thus self-adjoint) [1]. The definition can be generalized to include mixed states on $H_N$ as well.

Now recall that we can write a one-body operator on $H_N$ as follows:

$$ O = \sum\limits_{i,j\in J} \langle f_i,o\, f_j\rangle_{\mathfrak h}\,a^\dagger(f_i)\, a(f_j) \tag{2} \quad ,$$

which in turn means that we can compute the expectation value of $O$ in the state $\psi$ with

$$ \langle \psi, O\,\psi\rangle_{H_N} = \mathrm{Tr}_{\mathfrak h}\, \gamma_\psi\, o \quad .\tag{3}$$

This means that, at least in principle, we don't need the full wave function $\psi \in H_N$ to compute the expectation values of one-body operators; the corresponding 1-RDM $\gamma_\psi$ suffices.

To proceed, consider a Hamiltonian $H$ on $H_N$ consisting of the sum of a one and two-body operator, which we'll write as $H=T+ W$. An example is the Hamiltonian of $N$ electrons in a solid in the Born-Oppenheimer approximation, where $T$ is the kinetic energy plus external potential and $W$ the Coulomb interaction. It is very-well known that the (here assumed to be) non-degenerate ground state of $H$ is in one-to-one relation with its corresponding 1-RDM, which we will discuss in detail now.

The situation is very similar to the density functional case: Indeed, note that the density operator is a one-body operator and that given $\gamma_\psi$, we can obtain the density of $\psi$ from it. So for a local external potential and a non-degenerate ground state, the ground state density is in one-to-one correspondence with the ground state (Hohenberg-Kohn) and thus with the ground state 1-RDM. So the Hohenberg-Kohn theorem immediately applies to the 1-RDM too, at least in the case of local external potentials. However, as shown by Gilbert [2], the ground state 1-RDM also is in one-to-one correspondence with the ground state if the external potential is non-local. Yet, in general, there is no one-to-one correspondence between the ground state 1-RDM and the external potential anymore.

Similarly as in density functional theory (DFT), this means that all ground state properties are functionals of the ground state 1-RDM. This in particular includes the ground state energy, for which we can also formulate a variational principle. Let $\mathcal V$ denote the set of $v$-representable 1-RDMs$^{2}$. Then the ground state energy can be found as follows:

$$E_0 =\min\limits_{\gamma \in \mathcal V}\, \mathrm{Tr}_{\mathfrak h} \gamma_ \, t + W[\gamma] \tag{4} \quad ,$$

where $$W[\gamma] := \langle \psi[\gamma],W\,\psi[\gamma]\rangle_{H_N} \tag{5}$$ is the so-called universal functional and $\psi[\gamma]$ is the ground state associated to the ground state 1-RDM $\gamma$. However, again similar to the DFT case, the problem is that we can't efficiently characterize the set $\mathcal V$. To circumvent this problem, we can also formulate a Levy-Lieb constrained search [3,4]. For if $\mathcal P$ denotes the set of pure state $N$-representable 1-RDMs, then $$E_0 = \min\limits_{\gamma \in \mathcal P}\, \mathrm{Tr}_{\mathfrak h} \gamma_ \, t + F[\gamma] \tag{6} \quad , $$ where

$$ F[\gamma] : = \min\limits_{\psi \in H_N | \gamma_\psi = \gamma}\, \langle \psi, W\,\psi\rangle_{H_N} \quad . \tag{7}$$

Note that this construction removes the problems regarding degeneracies and $v$-representability. As briefly stated in the footnotes, the (pure state) $N$-representability problem can be solved in principle, i.e. we know the set $\mathcal P$.


$^1$ Most of the discussion can readily be applied to identical bosons, too [9-11]. Furthermore, Bose-Einstein condensation is defined in terms of the 1-RDM.

$^2$ Here it is assumed that the reader is familiar with the notion of $v$- and $N$-representability. It might be of interest to note that the fermionic pure state $N$-representability problem was solved only (more or less) recently, see [5,6]. The bosonic pure state $N$-representability problem is trivial. For both cases, the ensemble $N$-representability problems are well-known, cf. [7-9]. In contrast, the $v$-representability problem remains unknown (with a few exceptions).


References and further reading:

[1] Solovej, J. P. (2007). Many body quantum mechanics. Lecture Notes.

[2] Gilbert, T. L. (1975). Hohenberg-Kohn theorem for nonlocal external potentials. Physical Review B, 12(6), 2111.

[3] Levy, M. (1979). Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the v-representability problem. Proceedings of the National Academy of Sciences, 76(12), 6062-6065.

[4] Lieb, E. H. (2002). Density functionals for Coulomb systems. In Inequalities (pp. 269-303). Springer, Berlin, Heidelberg.

[5] Klyachko, A. A. (2006, April). Quantum marginal problem and N-representability. In Journal of Physics: Conference Series (Vol. 36, No. 1, p. 014). IOP Publishing.

[6] Altunbulak, M., & Klyachko, A. (2008). The Pauli principle revisited. Communications in Mathematical Physics, 282(2), 287-322.

[7] Coleman, A. J. (1963). Structure of fermion density matrices. Reviews of modern Physics, 35(3), 668.

[8] Valone, S. M. (1980). Consequences of extending 1‐matrix energy functionals from pure–state representable to all ensemble representable 1 matrices. The Journal of Chemical Physics, 73(3), 1344-1349.

[9] Giesbertz, K. J., & Ruggenthaler, M. (2019). One-body reduced density-matrix functional theory in finite basis sets at elevated temperatures. Physics Reports, 806, 1-47.

[10] Benavides-Riveros, C. L., Wolff, J., Marques, M. A., & Schilling, C. (2020). Reduced density matrix functional theory for bosons. Physical Review Letters, 124(18), 180603.

[11] Liebert, J., & Schilling, C. (2021). Functional theory for Bose-Einstein condensates. Physical Review Research, 3(1), 013282.

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