files / problems / simon-ionization-excess / compute / q1 / certs / hminus.json

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{
  "system": "H-",
  "Z": 1,
  "N": 2,
  "threshold": "-1/2",
  "strict": true,
  "is_new_bound": false,
  "note": "Variational replay that H- binds. Combined with Lieb Nc < 2Z+1 this recovers N0(1)=2, already in Lieb 1984. Not a new bound.",
  "primary": {
    "family": "Hylleraas",
    "psi": "exp(-alpha (r1+r2)) (1 + c r12)",
    "alpha": {
      "exact": "5/6",
      "float": 0.8333333333333334,
      "enclosure": [
        "0.8333333333333333333333333333333333333333",
        "0.8333333333333333333333333333333333333334"
      ]
    },
    "c": {
      "exact": "1/2",
      "float": 0.5,
      "enclosure": [
        "0.5000000000000000000000000000000000000000",
        "0.5000000000000000000000000000000000000000"
      ]
    },
    "energy": {
      "exact": "-815/1602",
      "float": -0.5087390761548065,
      "enclosure": [
        "-0.5087390761548064918851435705368289637953",
        "-0.5087390761548064918851435705368289637952"
      ]
    },
    "gap_below_threshold": {
      "exact": "7/801",
      "float": 0.008739076154806492,
      "enclosure": [
        "0.0087390761548064918851435705368289637952",
        "0.0087390761548064918851435705368289637953"
      ]
    },
    "T_over_N": {
      "exact": "21575/41652",
      "float": 0.5179823297800826,
      "enclosure": [
        "0.5179823297800825890713531162969365216556",
        "0.5179823297800825890713531162969365216557"
      ]
    },
    "V_over_N": {
      "exact": "-14255/13884",
      "float": -1.026721405934889,
      "enclosure": [
        "-1.0267214059348890809564966868337654854509",
        "-1.0267214059348890809564966868337654854508"
      ]
    }
  },
  "chandrasekhar": {
    "family": "Chandrasekhar",
    "psi": "exp(-a r1 - b r2) + exp(-b r1 - a r2)",
    "a": {
      "exact": "26/25",
      "float": 1.04,
      "enclosure": [
        "1.0400000000000000000000000000000000000000",
        "1.0400000000000000000000000000000000000000"
      ]
    },
    "b": {
      "exact": "7/25",
      "float": 0.28,
      "enclosure": [
        "0.2800000000000000000000000000000000000000",
        "0.2800000000000000000000000000000000000000"
      ]
    },
    "energy": {
      "exact": "-1076189452297/2096620401250",
      "float": -0.5132972338032095,
      "enclosure": [
        "-0.5132972338032094210978717099326422477738",
        "-0.5132972338032094210978717099326422477737"
      ]
    },
    "gap_below_threshold": {
      "exact": "13939625836/1048310200625",
      "float": 0.013297233803209421,
      "enclosure": [
        "0.0132972338032094210978717099326422477737",
        "0.0132972338032094210978717099326422477738"
      ]
    },
    "T_over_N": {
      "exact": "1076755797653/2096620401250",
      "float": 0.5135673567857304,
      "enclosure": [
        "0.5135673567857304088130769828356405248443",
        "0.5135673567857304088130769828356405248444"
      ]
    },
    "V_over_N": {
      "exact": "-43058904999/41932408025",
      "float": -1.0268645905889398,
      "enclosure": [
        "-1.0268645905889398299109486927682827726181",
        "-1.0268645905889398299109486927682827726180"
      ]
    }
  },
  "chandrasekhar_frozen_inner": {
    "comment": "a=1 frozen, b=2/7; already binds (H\u00f8gaasen\u2013Richard\u2013Sorba)",
    "a": {
      "exact": "1/1",
      "float": 1.0,
      "enclosure": [
        "1.0000000000000000000000000000000000000000",
        "1.0000000000000000000000000000000000000000"
      ]
    },
    "b": {
      "exact": "2/7",
      "float": 0.2857142857142857,
      "enclosure": [
        "0.2857142857142857142857142857142857142857",
        "0.2857142857142857142857142857142857142858"
      ]
    },
    "energy": {
      "exact": "-35516701/69291586",
      "float": -0.5125687410301158,
      "enclosure": [
        "-0.5125687410301158354204794792833865860713",
        "-0.5125687410301158354204794792833865860712"
      ]
    },
    "gap_below_threshold": {
      "exact": "435454/34645793",
      "float": 0.012568741030115835,
      "enclosure": [
        "0.0125687410301158354204794792833865860712",
        "0.0125687410301158354204794792833865860713"
      ]
    }
  },
  "uncorrelated_does_not_bind": {
    "alpha": {
      "exact": "11/16",
      "float": 0.6875,
      "enclosure": [
        "0.6875000000000000000000000000000000000000",
        "0.6875000000000000000000000000000000000000"
      ]
    },
    "energy": {
      "exact": "-121/256",
      "float": -0.47265625,
      "enclosure": [
        "-0.4726562500000000000000000000000000000000",
        "-0.4726562500000000000000000000000000000000"
      ]
    },
    "comment": "E = -(11/16)^2 = -121/256 > -1/2"
  },
  "published_comparison": {
    "best_NR_Hminus": "-0.527751016544377...",
    "source": "Nakashima\u2013Nakatsuji, J. Chem. Phys. 127, 224104 (2007), doi:10.1063/1.2801981",
    "comment": "Published benchmark, not replayed here. Our trials sit above it."
  },
  "script_sha256": "167c3ce632825368f61d2236d597e4962b2cd409024f249c657dbfe1cd05c983",
  "replay": "python3 hylleraas.py"
}