Recent progresses in the variational reduced-density-matrix method - PowerPoint PPT Presentation

. . Recent progresses in the variational reduced-density-matrix method . . . . . 中田真秀 (NAKATA, Maho) maho@riken.jp http://accc.riken.jp/maho/ 理化学研究所 (RIKEN), Advanced Center for Computing and Communication The 50th Sanibel Symposium (February 24 - March 2, 2010) NAKATA, Maho (RIKEN, ACCC) Recent progresses in the variational reduced-density-matrix method Sanibel symposium 2010/2/25 1 / 34

Collaborators current and past 福田光浩 (Fukuda Mituhiro) Michael Overton 安田耕二 (Yasuda Koji) Zhengji Zhao Bastiaan J. Braams 中田和秀 (Nakata Kazuhide) Jerome K. Percus 江原正博 (Ehara Masahiro) 藤澤克樹 (Fujisawa Katsuki) 山下真 (Yamashita Makoto) 中辻博 (Nakatsuji Hiroshi) NAKATA, Maho (RIKEN, ACCC) Recent progresses in the variational reduced-density-matrix method Sanibel symposium 2010/2/25 2 / 34

Overview Introduction of the RDM method. Recent results. Some open problems. NAKATA, Maho (RIKEN, ACCC) Recent progresses in the variational reduced-density-matrix method Sanibel symposium 2010/2/25 3 / 34

Part 1 Introduction of the RDM method. NAKATA, Maho (RIKEN, ACCC) Recent progresses in the variational reduced-density-matrix method Sanibel symposium 2010/2/25 4 / 34

What is the RDM method in short? NAKATA, Maho (RIKEN, ACCC) Recent progresses in the variational reduced-density-matrix method Sanibel symposium 2010/2/25 5 / 34

What is the RDM method in short? The RDM method: 2-RDM as basic variable NAKATA, Maho (RIKEN, ACCC) Recent progresses in the variational reduced-density-matrix method Sanibel symposium 2010/2/25 5 / 34

What is the RDM method in short? The RDM method: 2-RDM as basic variable Γ i 1 i 2 2 � Ψ | a † i 1 a † j 1 j 2 = 1 i 2 a j 2 a j 1 | Ψ � NAKATA, Maho (RIKEN, ACCC) Recent progresses in the variational reduced-density-matrix method Sanibel symposium 2010/2/25 5 / 34

What is the RDM method in short? The RDM method: 2-RDM as basic variable Γ i 1 i 2 2 � Ψ | a † i 1 a † j 1 j 2 = 1 i 2 a j 2 a j 1 | Ψ � Equivalent to the Schr¨ odinger equation NAKATA, Maho (RIKEN, ACCC) Recent progresses in the variational reduced-density-matrix method Sanibel symposium 2010/2/25 5 / 34

What is the RDM method in short? The RDM method: 2-RDM as basic variable Γ i 1 i 2 2 � Ψ | a † i 1 a † j 1 j 2 = 1 i 2 a j 2 a j 1 | Ψ � Equivalent to the Schr¨ odinger equation Ground state energy: Minimize directly! NAKATA, Maho (RIKEN, ACCC) Recent progresses in the variational reduced-density-matrix method Sanibel symposium 2010/2/25 5 / 34

What is the RDM method in short? The RDM method: 2-RDM as basic variable Γ i 1 i 2 2 � Ψ | a † i 1 a † j 1 j 2 = 1 i 2 a j 2 a j 1 | Ψ � Equivalent to the Schr¨ odinger equation Ground state energy: Minimize directly! N -representability condition; the only one approximation NAKATA, Maho (RIKEN, ACCC) Recent progresses in the variational reduced-density-matrix method Sanibel symposium 2010/2/25 5 / 34

Our goal: doing chemistry from the first principle, faster calculation and deeper understanding ✞ ☎ Our target ✝ ✆ ab initio ...theoretically and practically good approximation faster method ...mathematically simpler deeper understanding...electronic structure NAKATA, Maho (RIKEN, ACCC) Recent progresses in the variational reduced-density-matrix method Sanibel symposium 2010/2/25 6 / 34

Our goal: doing chemistry from the first principle, faster calculation and deeper understanding ✞ ☎ Our target ✝ ✆ ab initio ...theoretically and practically good approximation faster method ...mathematically simpler deeper understanding...electronic structure NAKATA, Maho (RIKEN, ACCC) Recent progresses in the variational reduced-density-matrix method Sanibel symposium 2010/2/25 7 / 34

The ground state and energy calculation [Husimi 1940], [L¨ owdin 1954], [Mayer 1955], [Coulson 1960], [Rosina 1968] NAKATA, Maho (RIKEN, ACCC) Recent progresses in the variational reduced-density-matrix method Sanibel symposium 2010/2/25 8 / 34

The ground state and energy calculation [Husimi 1940], [L¨ owdin 1954], [Mayer 1955], [Coulson 1960], [Rosina 1968] i a j + 1 ∑ ∑ j a † w i 1 i 2 j 1 j 2 a † i 1 a † v i H = i 2 a j 2 a j 1 2 ij i 1 i 2 j 1 j 2 NAKATA, Maho (RIKEN, ACCC) Recent progresses in the variational reduced-density-matrix method Sanibel symposium 2010/2/25 8 / 34

The ground state and energy calculation [Husimi 1940], [L¨ owdin 1954], [Mayer 1955], [Coulson 1960], [Rosina 1968] i a j + 1 ∑ ∑ j a † w i 1 i 2 j 1 j 2 a † i 1 a † v i H = i 2 a j 2 a j 1 2 ij i 1 i 2 j 1 j 2 The ground state energy becomes... E g = min � Ψ | H | Ψ � i a j | Ψ � + 1 ∑ ∑ j � Ψ | a † w i 1 i 2 j 1 j 2 � Ψ | a † i 1 a † v i = min i 2 a j 2 a j 1 | Ψ � 2 ij i 1 i 2 j 1 j 2 ∑ ∑ v i j γ i w i 1 i 2 j 1 j 2 Γ i 1 i 2 = min { j + j 1 j 2 } i j i 1 i 2 j 1 j 2 NAKATA, Maho (RIKEN, ACCC) Recent progresses in the variational reduced-density-matrix method Sanibel symposium 2010/2/25 8 / 34

The ground state and energy calculation [Husimi 1940], [L¨ owdin 1954], [Mayer 1955], [Coulson 1960], [Rosina 1968] i a j + 1 ∑ ∑ j a † w i 1 i 2 j 1 j 2 a † i 1 a † v i H = i 2 a j 2 a j 1 2 ij i 1 i 2 j 1 j 2 The ground state energy becomes... E g = min � Ψ | H | Ψ � i a j | Ψ � + 1 ∑ ∑ j � Ψ | a † w i 1 i 2 j 1 j 2 � Ψ | a † i 1 a † v i = min i 2 a j 2 a j 1 | Ψ � 2 ij i 1 i 2 j 1 j 2 ∑ ∑ v i j γ i w i 1 i 2 j 1 j 2 Γ i 1 i 2 = min { j + j 1 j 2 } i j i 1 i 2 j 1 j 2 Definition of 1, 2-RDMs j 1 j 2 = 1 Γ i 1 i 2 2 � Ψ | a † i 1 a † γ i j = � Ψ | a † i 2 a j 2 a j 1 | Ψ � , i a j | Ψ � . NAKATA, Maho (RIKEN, ACCC) Recent progresses in the variational reduced-density-matrix method Sanibel symposium 2010/2/25 8 / 34

N -representability condition NAKATA, Maho (RIKEN, ACCC) Recent progresses in the variational reduced-density-matrix method Sanibel symposium 2010/2/25 9 / 34

N -representability condition [Mayers 1955], [Tredgold 1957]: Far lower than the exact one NAKATA, Maho (RIKEN, ACCC) Recent progresses in the variational reduced-density-matrix method Sanibel symposium 2010/2/25 9 / 34

N -representability condition [Mayers 1955], [Tredgold 1957]: Far lower than the exact one N -representability condition [Coleman 1963] ∑ ∑ w i 1 i 2 j 1 j 2 Γ i 1 i 2 v i j γ i E g = min P { j + j 1 j 2 } ij i 1 i 2 j 1 j 2 γ, Γ ∈ P should satisfy N -representability condition: Γ (12 | 1 ′ 2 ′ ) → Ψ (123 · · · N ) γ (1 | 1 ′ ) → Ψ (123 · · · N ) . ✞ ☎ Encodes two-body effects completely. Very compact. ✝ ✆ NAKATA, Maho (RIKEN, ACCC) Recent progresses in the variational reduced-density-matrix method Sanibel symposium 2010/2/25 9 / 34

Approximate N -representability condition Approximation (necessary) condition : where Physics and Chemistry are NAKATA, Maho (RIKEN, ACCC) Recent progresses in the variational reduced-density-matrix method Sanibel symposium 2010/2/25 10 / 34

Approximate N -representability condition Approximation (necessary) condition : where Physics and Chemistry are P , Q -condition, ensemble 1-RDM condition [Coleman 1963] NAKATA, Maho (RIKEN, ACCC) Recent progresses in the variational reduced-density-matrix method Sanibel symposium 2010/2/25 10 / 34

Approximate N -representability condition Approximation (necessary) condition : where Physics and Chemistry are P , Q -condition, ensemble 1-RDM condition [Coleman 1963] G -condition [Garrod and Percus 1964] NAKATA, Maho (RIKEN, ACCC) Recent progresses in the variational reduced-density-matrix method Sanibel symposium 2010/2/25 10 / 34

Approximate N -representability condition Approximation (necessary) condition : where Physics and Chemistry are P , Q -condition, ensemble 1-RDM condition [Coleman 1963] G -condition [Garrod and Percus 1964] k -th order approximation [Erdahl, Jin 2000] (aka k -positivity [Mazziotti Erdahl 2001]) NAKATA, Maho (RIKEN, ACCC) Recent progresses in the variational reduced-density-matrix method Sanibel symposium 2010/2/25 10 / 34

Approximate N -representability condition Approximation (necessary) condition : where Physics and Chemistry are P , Q -condition, ensemble 1-RDM condition [Coleman 1963] G -condition [Garrod and Percus 1964] k -th order approximation [Erdahl, Jin 2000] (aka k -positivity [Mazziotti Erdahl 2001]) T 1 , T 2 , T 2 ′ , ( ¯ T 2 )-condition [Zhao et al. 2004], [Erdahl 1978] [Braams et al 2007] [Mazziotti 2006, 2007] NAKATA, Maho (RIKEN, ACCC) Recent progresses in the variational reduced-density-matrix method Sanibel symposium 2010/2/25 10 / 34

Approximate N -representability condition Approximation (necessary) condition : where Physics and Chemistry are P , Q -condition, ensemble 1-RDM condition [Coleman 1963] G -condition [Garrod and Percus 1964] k -th order approximation [Erdahl, Jin 2000] (aka k -positivity [Mazziotti Erdahl 2001]) T 1 , T 2 , T 2 ′ , ( ¯ T 2 )-condition [Zhao et al. 2004], [Erdahl 1978] [Braams et al 2007] [Mazziotti 2006, 2007] Davidson’s inequality [Davidson 1969][Ayers et al. 2006] NAKATA, Maho (RIKEN, ACCC) Recent progresses in the variational reduced-density-matrix method Sanibel symposium 2010/2/25 10 / 34