{
  "id": "effective-nuclear-charge",
  "version": "da1abed9eef1",
  "status": "published",
  "name": "Effective Nuclear Charge Calculator",
  "question": "What is the effective nuclear charge?",
  "summary": "Finds the effective nuclear charge Zeff = Z − S felt by an electron of any element, with the shielding constant S worked out by Slater’s rules from the element’s ground-state electron configuration.",
  "category": "chemistry",
  "subcategory": "atoms",
  "url": "https://www.acalculator.org/chemistry/effective-nuclear-charge-calculator",
  "markdown": "https://www.acalculator.org/chemistry/effective-nuclear-charge-calculator.md",
  "kind": "function",
  "method": "Zeff = Z − S. Group the electrons [1s] [2s,2p] [3s,3p] [3d] [4s,4p] [4d] [4f] [5s,5p] [5d] [5f] [6s,6p] [6d] [7s,7p]. For an s or p electron: 0.35 for each other electron in its group (0.30 in 1s), 0.85 for each in shell n − 1, 1.00 for each further in. For a d or f electron: 0.35 for each other in its group, 1.00 for each in a group to its left. Groups to the right add nothing.",
  "assumptions": [
    "Neutral atoms in their ground state: NIST’s configuration for Z up to 108, the Madelung filling order beyond.",
    "Slater’s rules are an approximation; they give one Zeff for s and p electrons of the same shell.",
    "The outermost electron is one in the last occupied group of the list (4s for iron, 4d for palladium)."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "z": {
        "title": "Element",
        "description": "The element, by atomic number, symbol and name.",
        "type": "string",
        "enum": [
          "1",
          "2",
          "3",
          "4",
          "5",
          "6",
          "7",
          "8",
          "9",
          "10",
          "11",
          "12",
          "13",
          "14",
          "15",
          "16",
          "17",
          "18",
          "19",
          "20",
          "21",
          "22",
          "23",
          "24",
          "25",
          "26",
          "27",
          "28",
          "29",
          "30",
          "31",
          "32",
          "33",
          "34",
          "35",
          "36",
          "37",
          "38",
          "39",
          "40",
          "41",
          "42",
          "43",
          "44",
          "45",
          "46",
          "47",
          "48",
          "49",
          "50",
          "51",
          "52",
          "53",
          "54",
          "55",
          "56",
          "57",
          "58",
          "59",
          "60",
          "61",
          "62",
          "63",
          "64",
          "65",
          "66",
          "67",
          "68",
          "69",
          "70",
          "71",
          "72",
          "73",
          "74",
          "75",
          "76",
          "77",
          "78",
          "79",
          "80",
          "81",
          "82",
          "83",
          "84",
          "85",
          "86",
          "87",
          "88",
          "89",
          "90",
          "91",
          "92",
          "93",
          "94",
          "95",
          "96",
          "97",
          "98",
          "99",
          "100",
          "101",
          "102",
          "103",
          "104",
          "105",
          "106",
          "107",
          "108",
          "109",
          "110",
          "111",
          "112",
          "113",
          "114",
          "115",
          "116",
          "117",
          "118"
        ]
      },
      "e": {
        "title": "Electron in",
        "description": "Which electron: the outermost one, or one in a chosen Slater group.",
        "type": "string",
        "enum": [
          "outer",
          "1s",
          "2sp",
          "3sp",
          "3d",
          "4sp",
          "4d",
          "4f",
          "5sp",
          "5d",
          "5f",
          "6sp",
          "6d",
          "7sp"
        ]
      }
    }
  },
  "outputs": {
    "zeff": {
      "label": "Effective nuclear charge (Zeff)",
      "description": "Z minus the shielding constant, felt by the chosen electron.",
      "format": "number"
    },
    "shielding": {
      "label": "Shielding constant (S)",
      "description": "The sum of the other electrons’ shielding by Slater’s rules.",
      "format": "number"
    },
    "z": {
      "label": "Nuclear charge (Z)",
      "description": "The atomic number.",
      "format": "integer"
    },
    "working": {
      "label": "Working",
      "description": "The shielding sum by kind of electron, then Z − S.",
      "format": "text"
    },
    "groups": {
      "label": "Slater groups",
      "description": "The ground-state configuration written in Slater’s groups.",
      "format": "text"
    },
    "which": {
      "label": "Electron",
      "description": "The group of the chosen electron and its element.",
      "format": "text"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "z": "26",
      "e": "outer"
    },
    "outputs": {
      "zeff": 3.75,
      "shielding": 22.25,
      "z": 26,
      "working": "S = 0.35 × 1 + 0.85 × 14 + 1.00 × 10 = 22.25; Zeff = 26 − 22.25 = 3.75",
      "groups": "(1s²) (2s² 2p⁶) (3s² 3p⁶) (3d⁶) (4s²)",
      "which": "a 4s electron of iron (Fe)"
    },
    "text": "By Slater’s rules, a 4s electron of iron (Fe) feels Zeff = 3.75."
  },
  "examples": [
    {
      "given": {
        "z": "26",
        "e": "outer"
      },
      "expect": {
        "shielding": 22.25,
        "zeff": 3.75,
        "working": "S = 0.35 × 1 + 0.85 × 14 + 1.00 × 10 = 22.25; Zeff = 26 − 22.25 = 3.75"
      },
      "source": "Wikipedia, Slater’s rules (Slater, Phys. Rev. 36, 57 (1930), https://doi.org/10.1103/PhysRev.36.57; the iron example), https://en.wikipedia.org/wiki/Slater%27s_rules: Fe 4s, S = 0.35 × 1 + 0.85 × 14 + 1.00 × 10 = 22.25, Zeff = 3.75"
    },
    {
      "given": {
        "z": "26",
        "e": "3d"
      },
      "expect": {
        "shielding": 19.75,
        "zeff": 6.25
      },
      "source": "Wikipedia, Slater’s rules (Slater, Phys. Rev. 36, 57 (1930), https://doi.org/10.1103/PhysRev.36.57; the iron example), https://en.wikipedia.org/wiki/Slater%27s_rules: Fe 3d, S = 0.35 × 5 + 1.00 × 18 = 19.75, Zeff = 6.25"
    },
    {
      "given": {
        "z": "26",
        "e": "3sp"
      },
      "expect": {
        "shielding": 11.25,
        "zeff": 14.75
      },
      "source": "Wikipedia, Slater’s rules (Slater, Phys. Rev. 36, 57 (1930), https://doi.org/10.1103/PhysRev.36.57; the iron example), https://en.wikipedia.org/wiki/Slater%27s_rules: Fe 3s and 3p, S = 0.35 × 7 + 0.85 × 8 + 1.00 × 2 = 11.25, Zeff = 14.75"
    },
    {
      "given": {
        "z": "26",
        "e": "1s"
      },
      "expect": {
        "shielding": 0.3,
        "zeff": 25.7
      },
      "source": "Wikipedia, Slater’s rules (Slater, Phys. Rev. 36, 57 (1930), https://doi.org/10.1103/PhysRev.36.57; the iron example), https://en.wikipedia.org/wiki/Slater%27s_rules: Fe 1s, S = 0.30, Zeff = 25.70"
    },
    {
      "given": {
        "z": "7",
        "e": "outer"
      },
      "expect": {
        "shielding": 3.1,
        "zeff": 3.9
      },
      "source": "LibreTexts, McCollum, 1.3.4 Slater’s Rules (Ursinus College CHEM322), https://chem.libretexts.org/Courses/Ursinus_College/CHEM322%3A_Inorganic_Chemistry/01%3A_Atomic_Structure/1.03%3A_Multi-Electron_Atoms/1.3.04%3A_Slater%27s_Rules: N 2p, S = 0.85(2) + 0.35(4) = 3.10; hand calculation in content.mdx: Zeff = 7 − 3.10 = 3.90"
    },
    {
      "given": {
        "z": "29",
        "e": "3d"
      },
      "expect": {
        "shielding": 21.15,
        "zeff": 7.85
      },
      "source": "LibreTexts, McCollum, 1.3.4 Slater’s Rules (Ursinus College CHEM322), https://chem.libretexts.org/Courses/Ursinus_College/CHEM322%3A_Inorganic_Chemistry/01%3A_Atomic_Structure/1.03%3A_Multi-Electron_Atoms/1.3.04%3A_Slater%27s_Rules: Cu 3d, S = 21.15, Zeff = 7.85"
    }
  ],
  "sources": [
    "J. C. Slater, Atomic Shielding Constants, Physical Review 36, 57–64 (1930): the original paper, at the publisher. https://doi.org/10.1103/PhysRev.36.57",
    "Wikipedia, Slater’s rules: the groups, the 0.35 (0.30 for 1s), 0.85 and 1.00 contributions, and the iron example (4s 3.75, 3d 6.25, 3s and 3p 14.75, 2s and 2p 21.85, 1s 25.70). https://en.wikipedia.org/wiki/Slater%27s_rules (retrieved 2026-10-19)",
    "LibreTexts, Brett McCollum, 1.3.4 Slater’s Rules (Ursinus College CHEM322: Inorganic Chemistry, CC BY-SA): Zeff = Z − S, the nitrogen 2p example (S = 3.10) and the copper 3d example (Zeff = 7.85). https://chem.libretexts.org/Courses/Ursinus_College/CHEM322%3A_Inorganic_Chemistry/01%3A_Atomic_Structure/1.03%3A_Multi-Electron_Atoms/1.3.04%3A_Slater%27s_Rules (retrieved 2026-10-19)",
    "OpenStax, Chemistry 2e, §6.5 Periodic Variations in Element Properties: Zeff = Z − shielding, and hydrogen’s Zeff equals Z. https://openstax.org/books/chemistry-2e/pages/6-5-periodic-variations-in-element-properties (retrieved 2026-10-19)",
    "NIST Standard Reference Database 111, Ground Levels and Ionization Energies for the Neutral Atoms: ground-state configurations for Z = 1 to 108, in its HTML table in the NIST Atomic Spectra Database. https://www.nist.gov/pml/ground-levels-and-ionization-energies-neutral-atoms (retrieved 2026-10-19)"
  ],
  "related": [
    "electron-configuration",
    "valence-electrons",
    "protons-neutrons-electrons",
    "electronegativity"
  ],
  "changelog": []
}