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 \u72b6\u614b i, j \u306b\u5bfe\u3059\u308b\u30b9\u30d4\u30f3\u306b\u95a2\u3059\u308b\u6f14\u7b97\u5b50\u3092\u305d\u308c\u305e\u308c\u3001Si\u3001Sj\u3068\u3059\u308b\u3068\u3001\u22122JSi\u22c5Sj{displaystyle -2J{boldsymbol {S}}_{i}cdot {boldsymbol {S}}_{j}}\u306e\u5f62\u3067\u8868\u3055\u308c\u308b\u76f8\u4e92\u4f5c\u7528\u304c\u4ea4\u63db\u76f8\u4e92\u4f5c\u7528\u3067\u3042\u308b\u3002J\u306f\u4ea4\u63db\u7a4d\u5206\u3068\u8a00\u3044\u3001\u5f8c\u3067\u8a73\u8ff0\u3059\u308b\u3002  \u30cf\u30fc\u30c8\u30ea\u30fc\u30fb\u30d5\u30a9\u30c3\u30af\u8fd1\u4f3c\u3067\u306e\u4ea4\u63db\u76f8\u4e92\u4f5c\u7528[\u7de8\u96c6]\u6700\u3082\u7c21\u5358\u306a\u5834\u5408\u3068\u3057\u3066\u30012\u500b\u306e\u96fb\u5b50\u304b\u3089\u306a\u308b\u7cfb\uff082\u96fb\u5b50\u7cfb\uff09\u3092\u8003\u3048\u308b\u3002\u96fb\u5b50\u306f\u534a\u6574\u6570\u306e\u30b9\u30d4\u30f3\u3092\u6301\u3064\u30d5\u30a7\u30eb\u30df\u7c92\u5b50\u306a\u306e\u3067\u3001\u3053\u308c\u306f2\u30b9\u30d4\u30f3\u7cfb\u3068\u8003\u3048\u308b\u3053\u3068\u3082\u3067\u304d\u308b\u3002\u307e\u305f\u3001\u30d5\u30a7\u30eb\u30df\u7c92\u5b50\u306a\u306e\u3067\u30d5\u30a7\u30eb\u30df\u7d71\u8a08\u306b\u5f93\u3044\u3001\u30d1\u30a6\u30ea\u306e\u6392\u4ed6\u539f\u7406\u306b\u3088\u308a\u30012\u3064\u306e\u96fb\u5b50\u304c\u540c\u3058\u72b6\u614b\u3092\u5360\u6709\u3059\u308b\u3053\u3068\u306f\u7981\u6b62\u3055\u308c\u308b\u3002\u307e\u305f\u3001\u5916\u5834\uff08\u539f\u5b50\u306e\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u306a\u3069\uff09\u3084\u30b9\u30d4\u30f3\u8ecc\u9053\u76f8\u4e92\u4f5c\u7528\u306a\u3069\u306f\u8003\u3048\u306a\u3044\u3053\u3068\u3068\u3059\u308b\u30022\u96fb\u5b50\u7cfb\u306b\u304a\u3051\u308b\u96fb\u5b50\u306e\u6ce2\u52d5\u95a2\u6570 \u03a8 \u306f\u30b9\u30ec\u30fc\u30bf\u30fc\u884c\u5217\u5f0f\u3067\u8868\u3059\u3068\u3001\u03a8=12[\u03c81(x1)\u03c82(x2)\u2212\u03c81(x2)\u03c82(x1)]{displaystyle Psi ={1 over {sqrt {2}}}[psi _{1}({boldsymbol {x}}_{1})psi _{2}({boldsymbol {x}}_{2})-psi _{1}({boldsymbol {x}}_{2})psi _{2}({boldsymbol {x}}_{1})]}\u3068\u306a\u308b\u3002\u03c81\u3001\u03c82 \u306f2\u96fb\u5b50\u7cfb\u306e\u305d\u308c\u305e\u308c\u306e\u96fb\u5b50\u306b\u5bfe\u5fdc\u3059\u308b\u6ce2\u52d5\u95a2\u6570\u3067\u3001\u3053\u308c\u306f\u5ea7\u6a19 r \u306b\u95a2\u3059\u308b\u90e8\u5206\uff08\u8ecc\u9053\u95a2\u6570\uff09\u3068\u30b9\u30d4\u30f3 \u03c3 \u306b\u95a2\u3059\u308b\u90e8\u5206\uff08\u30b9\u30d4\u30f3\u95a2\u6570\uff09\u3068\u306b\u5909\u6570\u5206\u96e2\u3067\u304d\u308b\uff08\u5ea7\u6a19 x \u306f\u3001x = (r, \u03c3) \u3067\u3042\u308b\uff09\u3002\u3053\u306e2\u96fb\u5b50\u7cfb\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u56fa\u6709\u5024\u554f\u984c\u3092\u89e3\u304f\u3068\u30012\u96fb\u5b50\u7cfb\uff08\u30b9\u30d4\u30f3\u3092\u305d\u308c\u305e\u308c s1\u3001s2\u3068\u3059\u308b\uff09\u306e\u56fa\u6709\u72b6\u614b\u3068\u3057\u3066\u30b9\u30d4\u30f3\u4e00\u91cd\u9805 (s1 + s2 = 0) \u3068\u30b9\u30d4\u30f3\u4e09\u91cd\u9805 (s1 + s2 = 1) \u3068\u3044\u30462\u3064\u306e\u72b6\u614b\u304c\u51fa\u3066\u304f\u308b\u3002\u30b9\u30d4\u30f3\u4e00\u91cd\u9805\u3067\u306f\u8ecc\u9053\u95a2\u6570\u90e8\u5206\u304c\u5ea7\u6a19\u306e\u7f6e\u63db\u306b\u5bfe\u3057\u3066\u5bfe\u79f0\u3067\u3001\u30b9\u30d4\u30f3\u95a2\u6570\u304c\u53cd\u5bfe\u79f0\u3068\u306a\u308a\u3001\u30b9\u30d4\u30f3\u4e09\u91cd\u9805\u3067\u306f\u305d\u306e\u9006\u3068\u306a\u308b\u3002\u3053\u306e\u3053\u3068\u304b\u3089\u30b9\u30d4\u30f3\u4e00\u91cd\u9805\u72b6\u614b\u3068\u30b9\u30d4\u30f3\u4e09\u91cd\u9805\u72b6\u614b\u3068\u3067\u7cfb\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u306b\u5dee\u304c\u751f\u3058\u308b\u3002\u3053\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u5dee\u3092\u5f15\u304d\u8d77\u3053\u3059\u306e\u304c\u4ea4\u63db\u76f8\u4e92\u4f5c\u7528\u3067\u3042\u308b\u3002\u8ecc\u9053\u95a2\u6570\u3092 \u03c6(r) \u3068\u3057\u3066\u3001\u30b9\u30d4\u30f3\u4e00\u91cd\u9805\u306e\u5834\u5408\u4e0a\u306e\u5f0f\u3067\u306e\u8ecc\u9053\u95a2\u6570\u90e8\u5206\u306f\u3001\u03a8singlet(r1,r2)=12[\u03d51(r1)\u03d52(r2)+\u03d51(r2)\u03d52(r1)]{displaystyle Psi _{mathrm {singlet} }({boldsymbol {r}}_{1},{boldsymbol {r}}_{2})={1 over {sqrt {2}}}[phi _{1}({boldsymbol {r}}_{1})phi _{2}({boldsymbol {r}}_{2})+phi _{1}({boldsymbol {r}}_{2})phi _{2}({boldsymbol {r}}_{1})]}\u3068\u306a\u308a\u3001\u30b9\u30d4\u30f3\u4e09\u91cd\u9805\u3067\u306f\u3001\u03a8triplet(r1,r2)=12[\u03d51(r1)\u03d52(r2)\u2212\u03d51(r2)\u03d52(r1)]{displaystyle Psi _{mathrm {triplet} }({boldsymbol {r}}_{1},{boldsymbol {r}}_{2})={1 over {sqrt {2}}}[phi _{1}({boldsymbol {r}}_{1})phi _{2}({boldsymbol {r}}_{2})-phi _{1}({boldsymbol {r}}_{2})phi _{2}({boldsymbol {r}}_{1})]}\u3068\u306a\u308b\u3002\u7cfb\u306e\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3\u3092H\u3068\u3057\u3066\u3001\u3053\u3053\u3067\u3082\u5916\u5834\u3068\u3057\u3066\u306e\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u306f\u8003\u3048\u305a\u96fb\u5b50\u9593\u76f8\u4e92\u4f5c\u7528\u306e\u90e8\u5206\u306e\u307f\u306b\u7740\u76ee\u3059\u308c\u3070\u3001\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3\u306e\u671f\u5f85\u5024\u3001Etotal=\u27e8\u03a8|H|\u03a8\u27e9{displaystyle E_{mathrm {total} }=langle Psi |H|Psi rangle }\u306e\u4e2d\u8eab\u306b\u3001\u4ee5\u4e0b\u306e2\u3064\u306e\u91cd\u8981\u306a\u9805\u304c\u51fa\u3066\u304f\u308b\u3002K=\u222c\u03d51\u2217(r1)\u03d52\u2217(r2)e2\u03b1|r1\u2212r2|\u03d52(r2)\u03d51(r1)dr1dr2J=\u222c\u03d51\u2217(r1)\u03d52\u2217(r2)e2\u03b1|r1\u2212r2|\u03d52(r1)\u03d51(r2)dr1dr2{displaystyle {begin{aligned}K&=iint phi _{1}^{*}({boldsymbol {r}}_{1})phi _{2}^{*}({boldsymbol {r}}_{2}){e^{2} over {alpha |{boldsymbol {r}}_{1}-{boldsymbol {r}}_{2}|}}phi _{2}({boldsymbol {r}}_{2})phi _{1}({boldsymbol {r}}_{1})d{boldsymbol {r}}_{1}d{boldsymbol {r}}_{2}J&=iint phi _{1}^{*}({boldsymbol {r}}_{1})phi _{2}^{*}({boldsymbol {r}}_{2}){e^{2} over {alpha |{boldsymbol {r}}_{1}-{boldsymbol {r}}_{2}|}}phi _{2}({boldsymbol {r}}_{1})phi _{1}({boldsymbol {r}}_{2})d{boldsymbol {r}}_{1}d{boldsymbol {r}}_{2}end{aligned}}}K \u3092\u30af\u30fc\u30ed\u30f3\u7a4d\u5206\u3001J \u3092\u4ea4\u63db\u7a4d\u5206\u3068\u8a00\u3046\u3002\u5f8c\u8005\u306f\u3001\u96fb\u5b50\u306e\u5ea7\u6a19\u306e\u4ea4\u63db\u306b\u3088\u3063\u3066\u51fa\u3066\u304f\u308b\u9805\u306e\u305f\u3081\u3053\u306e\u540d\u304c\u4ed8\u3044\u3066\u3044\u308b\u3002\u03b1 = 4\u03c0\u03b50 \u3067\u3001 \u03b50 \u306f\u771f\u7a7a\u306e\u8a98\u96fb\u7387\u3067\u3042\u308b\u3002\u30b9\u30d4\u30f3\u4e00\u91cd\u9805\u306b\u5bfe\u5fdc\u3059\u308b\u56fa\u6709\u30a8\u30cd\u30eb\u30ae\u30fc\u90e8\u5206\u306f\u3001Esinglet=K+J{displaystyle E_{mathrm {singlet} }=K+J}\u30b9\u30d4\u30f3\u4e09\u91cd\u9805\u306b\u5bfe\u5fdc\u3059\u308b\u56fa\u6709\u30a8\u30cd\u30eb\u30ae\u30fc\u90e8\u5206\u306f\u3001Etriplet=K\u2212J{displaystyle E_{mathrm {triplet} }=K-J}\u3068\u306a\u308b\u3002\u3053\u308c\u304b\u3089\u4ea4\u63db\u7a4d\u5206\u304c\u6b63\u306e\u5834\u5408\u3001\u30a8\u30cd\u30eb\u30ae\u30fc\u3068\u3057\u3066 Etriplet \u306e\u65b9\u304c\u4f4e\u304f\u306a\u308a\u3001\u30b9\u30d4\u30f3\u4e09\u91cd\u9805\uff082\u3064\u306e\u30b9\u30d4\u30f3\u304c\u5e73\u884c\uff09\u306e\u65b9\u304c\u3088\u308a\u30a8\u30cd\u30eb\u30ae\u30fc\u7684\u306b\u5b89\u5b9a\u3068\u306a\u308b\u3002\u9006\u306b\u4ea4\u63db\u7a4d\u5206\u304c\u8ca0\u306e\u5834\u5408\u3001Esinglet \u306e\u65b9\u304c\u4f4e\u304f\u306a\u308a\u3001\u30b9\u30d4\u30f3\u4e00\u91cd\u9805\uff082\u3064\u306e\u30b9\u30d4\u30f3\u304c\u53cd\u5e73\u884c\uff09\u306e\u65b9\u304c\u3088\u308a\u5b89\u5b9a\u3068\u306a\u308b\u3002\u3053\u306e\u3088\u3046\u306b\u4ea4\u63db\u7a4d\u5206\u306e\u6b63\u8ca0\u306b\u3088\u308a\u30b9\u30d4\u30f3\u4e00\u91cd\u9805\u3001\u4e09\u91cd\u9805\u306e\u9593\u3067\u30a8\u30cd\u30eb\u30ae\u30fc\u5dee\u3068\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u5927\u5c0f\u304c\u751f\u3058\u3001\u3053\u308c\u3092\u5f15\u304d\u8d77\u3053\u3059\u306e\u304c\u4ea4\u63db\u76f8\u4e92\u4f5c\u7528\u3068\u8a00\u3048\u308b\u3002\u30b9\u30d4\u30f3\u304c\u5e73\u884c\u3001\u53cd\u5e73\u884c\u3069\u3061\u3089\u304c\u3088\u308a\u5b89\u5b9a\u3067\u3042\u308b\u304b\u304c\u3001\u7cfb\u306e\u78c1\u6c17\u69cb\u9020\uff08\u5f37\u78c1\u6027\u304b\u53cd\u5f37\u78c1\u6027\u304b\uff09\u304c\u3069\u3046\u306a\u308b\u304b\u306b\u6df1\u304f\u95a2\u4fc2\u3059\u308b\u30022\u96fb\u5b50\u306e\u30b9\u30d4\u30f3\u3092 s1\u3001s2 \u3068\u3057\u3066\u5148\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u90e8\u5206 E \u3092\u8868\u3059\u3068\u3001E=K\u221212J[1+4(s1\u22c5s2)]{displaystyle E=K-{1 over 2}J[1+4({boldsymbol {s}}_{1}cdot {boldsymbol {s}}_{2})]}\u3068\u306a\u308b\u3002\u3053\u306e\u6642\u3001\u8ecc\u9053\u95a2\u6570 \u03c61(r) \u3068 \u03c62(r) \u306f\u4e92\u3044\u306b\u76f4\u4ea4\u3059\u308b\u5834\u5408\u3001\u4ea4\u63db\u7a4d\u5206 J 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