# Küresel harmoniklerin tablosu
Bu bir Küresel harmonikler ortonormalize tablosudur ve Bu Condon-Shortley fazı "l" = 10 dereceye kadar sağlanır. Bazen bu formüllerin "Kartezyen" yorumu verilir. Bu varsayım "x", "y", "z" ve "r" Kartezyen-e-küresel koordinat dönüşümü yoluyla θ ve φ ye ilişkindir:
 x amp; = r sinθcosφ
 y amp; = r sinθsinφ
 z amp; = r cosθ


## Küresel harmonikler.
Burada dikkat çekmesi gereken nokta iki değişkenli fonksiyonların bir yüzeye karşılık geldiğidir. Yani bu harmonikler küre içinde farklı farklı yüzeylerin dalgalanmaları olacaktır


### l = 0.
Y_0^0(θ,φ)=12√(1π)


### l = 1.
 
 Y_1^-1(θ,φ) amp; = 1 2√(3 2π)· e^-iφ·sinθ 
 = 1 2√(3 2π)·(x-iy) r
 Y_1^0(θ,φ) amp; = 1 2√(3π)·cosθ 
 = 1 2√(3π)·z r
 Y_1^1(θ,φ) amp; = -1 2√(3 2π)· e^iφ·sinθ 
 = -1 2√(3 2π)·(x+iy) r


### l = 2.
Y_2^-2(θ,φ)
=14√(152π)·e^-2iφ·sin^2θ =14√(152π)·(x - iy)^2 r^2
Y_2^-1(θ,φ)
=12√(152π)·e^-iφ·sinθ·cosθ =12√(152π)·(x - iy)z r^2
Y_2^0(θ,φ)
=14√(5π)·(3cos^2θ-1) =14√(5π)·(2z^2-x^2-y^2)r^2
Y_2^1(θ,φ)
=-12√(152π)·e^iφ·sinθ·cosθ =-12√(152π)·(x + iy)z r^2
Y_2^2(θ,φ)
=14√(152π)·e^2iφ·sin^2θ =14√(152π)·(x + iy)^2 r^2


### l = 3.
Y_3^-3(θ,φ)
= 18√(35π)·e^-3iφ·sin^3θ = 18√(35π)·(x - iy)^3r^3
Y_3^-2(θ,φ)
= 14√(1052π)·e^-2iφ·sin^2θ·cosθ = 14√(1052π)·(x- iy)^2 z r^3
Y_3^-1(θ,φ)
=18√(21π)·e^-iφ·sinθ·(5cos^2θ-1) =18√(21π)·(x - iy)(4z^2- x^2 - y^2)r^3
Y_3^0(θ,φ)
=14√(7π)·(5cos^3θ-3cosθ) =14√(7π)·z(2z^2 - 3x^2 - 3y^2)r^3
Y_3^1(θ,φ)
=-18√(21π)·e^iφ·sinθ·(5cos^2θ-1) =-18√(21π)·(x + iy) (4z^2 - x^2 - y^2) r^3
Y_3^2(θ,φ)
=14√(1052π)·e^2iφ·sin^2θ·cosθ =14√(1052π)·(x + iy)^2 z r^3
Y_3^3(θ,φ)
=-18√(35π)·e^3iφ·sin^3θ =-18√(35π)·(x + iy)^3r^3


### l = 4.
Y_4^-4(θ,φ)=316√(352π)·e^-4iφ·sin^4θ= 3/16 √(35/2 π) ·(x - i y)^4/r^4
Y_4^-3(θ,φ)=38√(35π)·e^-3iφ·sin^3θ·cosθ= 3/8 √(35/π) ·(x - i y)^3 z/r^4
Y_4^-2(θ,φ)=38√(52π)·e^-2iφ·sin^2θ·(7cos^2θ-1)
= 3/8 √(5/2 π) ·(x - i y)^2 ·(7 z^2 - r^2)/r^4
Y_4^-1(θ,φ)=38√(5π)·e^-iφ·sinθ·(7cos^3θ-3cosθ)
= 3/8 √(5/π) ·(x - i y) ·z ·(7 z^2 - 3 r^2)/r^4
Y_4^0(θ,φ)=316√(1π)·(35cos^4θ-30cos^2θ+3)
= 3/16 √(1/π) ·(35 z^4 - 30 z^2 r^2 + 3 r^4)/r^4
Y_4^1(θ,φ)=-38√(5π)·e^iφ·sinθ·(7cos^3θ-3cosθ)
= - 3/8 √(5/π) ·(x + i y) ·z ·(7 z^2 - 3 r^2)/r^4
Y_4^2(θ,φ)=38√(52π)·e^2iφ·sin^2θ·(7cos^2θ-1)
= 3/8 √(5/2 π) ·(x + i y)^2 ·(7 z^2 - r^2)/r^4
Y_4^3(θ,φ)=-38√(35π)·e^3iφ·sin^3θ·cosθ= - 3/8 √(35/π) ·(x + i y)^3 z/r^4
Y_4^4(θ,φ)=316√(352π)·e^4iφ·sin^4θ= 3/16 √(35/2 π) ·(x + i y)^4/r^4


### l = 5.
Y_5^-5(θ,φ)=332√(77π)·e^-5iφ·sin^5θ
Y_5^-4(θ,φ)=316√(3852π)·e^-4iφ·sin^4θ·cosθ
Y_5^-3(θ,φ)=132√(385π)·e^-3iφ·sin^3θ·(9cos^2θ-1)
Y_5^-2(θ,φ)=18√(11552π)·e^-2iφ·sin^2θ·(3cos^3θ-cosθ)
Y_5^-1(θ,φ)=116√(1652π)·e^-iφ·sinθ·(21cos^4θ-14cos^2θ+1)
Y_5^0(θ,φ)=116√(11π)·(63cos^5θ-70cos^3θ+15cosθ)
Y_5^1(θ,φ)=-116√(1652π)·e^iφ·sinθ·(21cos^4θ-14cos^2θ+1)
Y_5^2(θ,φ)=18√(11552π)·e^2iφ·sin^2θ·(3cos^3θ-cosθ)
Y_5^3(θ,φ)=-132√(385π)·e^3iφ·sin^3θ·(9cos^2θ-1)
Y_5^4(θ,φ)=316√(3852π)·e^4iφ·sin^4θ·cosθ
Y_5^5(θ,φ)=-332√(77π)·e^5iφ·sin^5θ


### l = 6.
Y_6^-6(θ,φ)=164√(3003π)·e^-6iφ·sin^6θ
Y_6^-5(θ,φ)=332√(1001π)·e^-5iφ·sin^5θ·cosθ
Y_6^-4(θ,φ)=332√(912π)·e^-4iφ·sin^4θ·(11cos^2θ-1)
Y_6^-3(θ,φ)=132√(1365π)·e^-3iφ·sin^3θ·(11cos^3θ-3cosθ)
Y_6^-2(θ,φ)=164√(1365π)·e^-2iφ·sin^2θ·(33cos^4θ-18cos^2θ+1)
Y_6^-1(θ,φ)=116√(2732π)·e^-iφ·sinθ·(33cos^5θ-30cos^3θ+5cosθ)
Y_6^0(θ,φ)=132√(13π)·(231cos^6θ-315cos^4θ+105cos^2θ-5)
Y_6^1(θ,φ)=-116√(2732π)·e^iφ·sinθ·(33cos^5θ-30cos^3θ+5cosθ)
Y_6^2(θ,φ)=164√(1365π)·e^2iφ·sin^2θ·(33cos^4θ-18cos^2θ+1)
Y_6^3(θ,φ)=-132√(1365π)·e^3iφ·sin^3θ·(11cos^3θ-3cosθ)
Y_6^4(θ,φ)=332√(912π)·e^4iφ·sin^4θ·(11cos^2θ-1)
Y_6^5(θ,φ)=-332√(1001π)·e^5iφ·sin^5θ·cosθ
Y_6^6(θ,φ)=164√(3003π)·e^6iφ·sin^6θ


### l = 7.
Y_7^-7(θ,φ)=364√(7152π)·e^-7iφ·sin^7θ
Y_7^-6(θ,φ)=364√(5005π)·e^-6iφ·sin^6θ·cosθ
Y_7^-5(θ,φ)=364√(3852π)·e^-5iφ·sin^5θ·(13cos^2θ-1)
Y_7^-4(θ,φ)=332√(3852π)·e^-4iφ·sin^4θ·(13cos^3θ-3cosθ)
Y_7^-3(θ,φ)=364√(352π)·e^-3iφ·sin^3θ·(143cos^4θ-66cos^2θ+3)
Y_7^-2(θ,φ)=364√(35π)·e^-2iφ·sin^2θ·(143cos^5θ-110cos^3θ+15cosθ)
Y_7^-1(θ,φ)=164√(1052π)·e^-iφ·sinθ·(429cos^6θ-495cos^4θ+135cos^2θ-5)
Y_7^0(θ,φ)=132√(15π)·(429cos^7θ-693cos^5θ+315cos^3θ-35cosθ)
Y_7^1(θ,φ)=-164√(1052π)·e^iφ·sinθ·(429cos^6θ-495cos^4θ+135cos^2θ-5)
Y_7^2(θ,φ)=364√(35π)·e^2iφ·sin^2θ·(143cos^5θ-110cos^3θ+15cosθ)
Y_7^3(θ,φ)=-364√(352π)·e^3iφ·sin^3θ·(143cos^4θ-66cos^2θ+3)
Y_7^4(θ,φ)=332√(3852π)·e^4iφ·sin^4θ·(13cos^3θ-3cosθ)
Y_7^5(θ,φ)=-364√(3852π)·e^5iφ·sin^5θ·(13cos^2θ-1)
Y_7^6(θ,φ)=364√(5005π)·e^6iφ·sin^6θ·cosθ
Y_7^7(θ,φ)=-364√(7152π)·e^7iφ·sin^7θ


### l = 8.
Y_8^-8(θ,φ)=3256√(121552π)·e^-8iφ·sin^8θ
Y_8^-7(θ,φ)=364√(121552π)·e^-7iφ·sin^7θ·cosθ
Y_8^-6(θ,φ)=1128√(7293π)·e^-6iφ·sin^6θ·(15cos^2θ-1)
Y_8^-5(θ,φ)=364√(170172π)·e^-5iφ·sin^5θ·(5cos^3θ-cosθ)
Y_8^-4(θ,φ)=3128√(13092π)·e^-4iφ·sin^4θ·(65cos^4θ-26cos^2θ+1)
Y_8^-3(θ,φ)=164√(196352π)·e^-3iφ·sin^3θ·(39cos^5θ-26cos^3θ+3cosθ)
Y_8^-2(θ,φ)=3128√(595π)·e^-2iφ·sin^2θ·(143cos^6θ-143cos^4θ+33cos^2θ-1)
Y_8^-1(θ,φ)=364√(172π)·e^-iφ·sinθ·(715cos^7θ-1001cos^5θ+385cos^3θ-35cosθ)
Y_8^0(θ,φ)=1256√(17π)·(6435cos^8θ-12012cos^6θ+6930cos^4θ-1260cos^2θ+35)
Y_8^1(θ,φ)=-364√(172π)·e^iφ·sinθ·(715cos^7θ-1001cos^5θ+385cos^3θ-35cosθ)
Y_8^2(θ,φ)=3128√(595π)·e^2iφ·sin^2θ·(143cos^6θ-143cos^4θ+33cos^2θ-1)
Y_8^3(θ,φ)=-164√(196352π)·e^3iφ·sin^3θ·(39cos^5θ-26cos^3θ+3cosθ)
Y_8^4(θ,φ)=3128√(13092π)·e^4iφ·sin^4θ·(65cos^4θ-26cos^2θ+1)
Y_8^5(θ,φ)=-364√(170172π)·e^5iφ·sin^5θ·(5cos^3θ-cosθ)
Y_8^6(θ,φ)=1128√(7293π)·e^6iφ·sin^6θ·(15cos^2θ-1)
Y_8^7(θ,φ)=-364√(121552π)·e^7iφ·sin^7θ·cosθ
Y_8^8(θ,φ)=3256√(121552π)·e^8iφ·sin^8θ


### l = 9.
Y_9^-9(θ,φ)=1512√(230945π)·e^-9iφ·sin^9θ
Y_9^-8(θ,φ)=3256√(2309452π)·e^-8iφ·sin^8θ·cosθ
Y_9^-7(θ,φ)=3512√(13585π)·e^-7iφ·sin^7θ·(17cos^2θ-1)
Y_9^-6(θ,φ)=1128√(40755π)·e^-6iφ·sin^6θ·(17cos^3θ-3cosθ)
Y_9^-5(θ,φ)=3256√(2717π)·e^-5iφ·sin^5θ·(85cos^4θ-30cos^2θ+1)
Y_9^-4(θ,φ)=3128√(950952π)·e^-4iφ·sin^4θ·(17cos^5θ-10cos^3θ+cosθ)
Y_9^-3(θ,φ)=1256√(21945π)·e^-3iφ·sin^3θ·(221cos^6θ-195cos^4θ+39cos^2θ-1)
Y_9^-2(θ,φ)=3128√(1045π)·e^-2iφ·sin^2θ·(221cos^7θ-273cos^5θ+91cos^3θ-7cosθ)
Y_9^-1(θ,φ)=3256√(952π)·e^-iφ·sinθ·(2431cos^8θ-4004cos^6θ+2002cos^4θ-308cos^2θ+7)
Y_9^0(θ,φ)=1256√(19π)·(12155cos^9θ-25740cos^7θ+18018cos^5θ-4620cos^3θ+315cosθ)
Y_9^1(θ,φ)=-3256√(952π)·e^iφ·sinθ·(2431cos^8θ-4004cos^6θ+2002cos^4θ-308cos^2θ+7)
Y_9^2(θ,φ)=3128√(1045π)·e^2iφ·sin^2θ·(221cos^7θ-273cos^5θ+91cos^3θ-7cosθ)
Y_9^3(θ,φ)=-1256√(21945π)·e^3iφ·sin^3θ·(221cos^6θ-195cos^4θ+39cos^2θ-1)
Y_9^4(θ,φ)=3128√(950952π)·e^4iφ·sin^4θ·(17cos^5θ-10cos^3θ+cosθ)
Y_9^5(θ,φ)=-3256√(2717π)·e^5iφ·sin^5θ·(85cos^4θ-30cos^2θ+1)
Y_9^6(θ,φ)=1128√(40755π)·e^6iφ·sin^6θ·(17cos^3θ-3cosθ)
Y_9^7(θ,φ)=-3512√(13585π)·e^7iφ·sin^7θ·(17cos^2θ-1)
Y_9^8(θ,φ)=3256√(2309452π)·e^8iφ·sin^8θ·cosθ
Y_9^9(θ,φ)=-1512√(230945π)·e^9iφ·sin^9θ


### l = 10.
Y_10^-10(θ,φ)=11024√(969969π)·e^-10iφ·sin^10θ
Y_10^-9(θ,φ)=1512√(4849845π)·e^-9iφ·sin^9θ·cosθ
Y_10^-8(θ,φ)=1512√(2552552π)·e^-8iφ·sin^8θ·(19cos^2θ-1)
Y_10^-7(θ,φ)=3512√(85085π)·e^-7iφ·sin^7θ·(19cos^3θ-3cosθ)
Y_10^-6(θ,φ)=31024√(5005π)·e^-6iφ·sin^6θ·(323cos^4θ-102cos^2θ+3)
Y_10^-5(θ,φ)=3256√(1001π)·e^-5iφ·sin^5θ·(323cos^5θ-170cos^3θ+15cosθ)
Y_10^-4(θ,φ)=3256√(50052π)·e^-4iφ·sin^4θ·(323cos^6θ-255cos^4θ+45cos^2θ-1)
Y_10^-3(θ,φ)=3256√(5005π)·e^-3iφ·sin^3θ·(323cos^7θ-357cos^5θ+105cos^3θ-7cosθ)
Y_10^-2(θ,φ)=3512√(3852π)·e^-2iφ·sin^2θ·(4199cos^8θ-6188cos^6θ+2730cos^4θ-364cos^2θ+7)
Y_10^-1(θ,φ)=1256√(11552π)·e^-iφ·sinθ·(4199cos^9θ-7956cos^7θ+4914cos^5θ-1092cos^3θ+63cosθ)
Y_10^0(θ,φ)=1512√(21π)·(46189cos^10θ-109395cos^8θ+90090cos^6θ-30030cos^4θ+3465cos^2θ-63)
Y_10^1(θ,φ)=-1256√(11552π)·e^iφ·sinθ·(4199cos^9θ-7956cos^7θ+4914cos^5θ-1092cos^3θ+63cosθ)
Y_10^2(θ,φ)=3512√(3852π)·e^2iφ·sin^2θ·(4199cos^8θ-6188cos^6θ+2730cos^4θ-364cos^2θ+7)
Y_10^3(θ,φ)=-3256√(5005π)·e^3iφ·sin^3θ·(323cos^7θ-357cos^5θ+105cos^3θ-7cosθ)
Y_10^4(θ,φ)=3256√(50052π)·e^4iφ·sin^4θ·(323cos^6θ-255cos^4θ+45cos^2θ-1)
Y_10^5(θ,φ)=-3256√(1001π)·e^5iφ·sin^5θ·(323cos^5θ-170cos^3θ+15cosθ)
Y_10^6(θ,φ)=31024√(5005π)·e^6iφ·sin^6θ·(323cos^4θ-102cos^2θ+3)
Y_10^7(θ,φ)=-3512√(85085π)·e^7iφ·sin^7θ·(19cos^3θ-3cosθ)
Y_10^8(θ,φ)=1512√(2552552π)·e^8iφ·sin^8θ·(19cos^2θ-1)
Y_10^9(θ,φ)=-1512√(4849845π)·e^9iφ·sin^9θ·cosθ
Y_10^10(θ,φ)=11024√(969969π)·e^10iφ·sin^10θ


## Gerçek küresel harmonikler.
Her Gerçek küresel harmonik için, karşılık gelen ("s", "p", "d", "f", "g") atomik orbital sembolü de bildirilmektedir.


### l = 0.
 Y_00 amp; = s = Y_0^0 = 1/2√(1/π)


### l = 1.
 
 Y_1,-1 amp; = p_y = i √(1/2)( Y_1^- 1 + Y_1^1 ) = √(3/4 π)·y/r
 Y_10 amp; = p_z = Y_1^0 = √(3/4 π)·z/r
 Y_11 amp; = p_x = √(1/2)( Y_1^- 1 - Y_1^1 ) = √(3/4 π)·x/r


### l = 2.
 Y_2,-2 amp; = d_xy = i √(1/2)( Y_2^- 2 - Y_2^2) = 1/2√(15/π)·x y/r^2
 Y_2,-1 amp; = d_yz = i √(1/2)( Y_2^- 1 + Y_2^1 ) = 1/2√(15/π)·y z/r^2
 Y_20 amp; = d_z^2 = Y_2^0 = 1/4√(5/π)·- x^2 - y^2 + 2 z^2/r^2
 Y_21 amp; = d_xz = √(1/2)( Y_2^- 1 - Y_2^1 ) = 1/2√(15/π)·z x/r^2
 Y_22 amp; = d_x^2-y^2 = √(1/2)( Y_2^- 2 + Y_2^2 ) = 1/4√(15/π)·x^2 - y^2 /r^2


### l = 3.
 Y_3,-3 amp; = f_y(3x^2-y^2) = i √(1/2)( Y_3^- 3 + Y_3^3 ) = 1/4√(35/2 π)·( 3 x^2 - y^2 ) y/r^3
 Y_3,-2 amp; = f_xyz = i √(1/2)( Y_3^- 2 - Y_3^2 ) = 1/2√(105/π)·xy z/r^3
 Y_3,-1 amp; = f_yz^2 = i √(1/2)( Y_3^- 1 + Y_3^1 ) = 1/4√(21/2 π)·y (4 z^2 - x^2 - y^2)/r^3
 Y_30 amp; = f_z^3 = Y_3^0 = 1/4√(7/π)·z (2 z^2 - 3 x^2 - 3 y^2)/r^3
 Y_31 amp; = f_xz^2 = √(1/2)( Y_3^- 1 - Y_3^1 ) = 1/4√(21/2 π)·x (4 z^2 - x^2 - y^2)/r^3
 Y_32 amp; = f_z(x^2-y^2) = √(1/2)( Y_3^- 2 + Y_3^2 ) = 1/4√(105/π)·( x^2 - y^2 ) z/r^3
 Y_33 amp; = f_x(x^2-3y^2) = √(1/2)( Y_3^- 3 - Y_3^3 ) = 1/4√(35/2 π)·( x^2 - 3 y^2 ) x/r^3


### l = 4.
 Y_4,-4 amp; = g_xy(x^2-y^2) = i √(1/2)( Y_4^- 4 - Y_4^4 ) = 3/4√(35/π)·xy ( x^2 - y^2 )/r^4
 Y_4,-3 amp; = g_zy^3 = i √(1/2)( Y_4^- 3 + Y_4^3 ) = 3/4√(35/2 π)·(3 x^2 - y^2) yz/r^4
 Y_4,-2 amp; = g_z^2xy = i √(1/2)( Y_4^- 2 - Y_4^2 ) = 3/4√(5/π)·xy · (7 z^2 - r^2)/r^4
 Y_4,-1 amp; = g_z^3y = i √(1/2)( Y_4^- 1 + Y_4^1) = 3/4√(5/2 π)·yz · (7 z^2 - 3 r^2)/r^4
 Y_40 amp; = g_z^4 = Y_4^0 = 3/16√(1/π)·(35 z^4 - 30 z^2 r^2 + 3 r^4)/r^4
 Y_41 amp; = g_z^3x = √(1/2)( Y_4^- 1 - Y_4^1 ) = 3/4√(5/2 π)·xz · (7 z^2 - 3 r^2)/r^4
 Y_42 amp; = g_z^2xy = √(1/2)( Y_4^- 2 + Y_4^2 ) = 3/8√(5/π)·(x^2 - y^2) · (7 z^2 - r^2)/r^4
 Y_43 amp; = g_zx^3 = √(1/2)( Y_4^- 3 - Y_4^3 ) = 3/4√(35/2 π)·(x^2 - 3 y^2) xz/r^4
 Y_44 amp; = g_x^4+y^4 = √(1/2)( Y_4^- 4 + Y_4^4 ) = 3/16√(35/π)·x^2 ( x^2 - 3 y^2 ) - y^2 ( 3 x^2 - y^2 )/r^4